From Coq Require Import ssreflect ssrfun ssrbool.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.

From stdpp Require Import base tactics sets.
New coercion path [Bool.Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]
New coercion path [Is_true] : bool >-> Sortclass is ambiguous with existing 
[is_true] : bool >-> Sortclass.
[ambiguous-paths,typechecker]

Require Import MatchingLogic.Utils.extralibrary. (* compare_nat *)

From MatchingLogic Require Import
    Pattern.

Section subst.
    Context {Σ : Signature}.

     (* There are two substitution operations over patterns, [bevar_subst] and [bsvar_subst]. *)
  (* substitute bound variable x for psi in phi *)
  Fixpoint bevar_subst (phi psi : Pattern) (x : db_index) :=
    match phi with
    | patt_free_evar x' => patt_free_evar x'
    | patt_free_svar x' => patt_free_svar x'
    | patt_bound_evar n => match compare_nat n x with
                           | Nat_less _ _ _ => patt_bound_evar n
                           | Nat_equal _ _ _ => psi
                           | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
                           end
    | patt_bound_svar n => patt_bound_svar n
    | patt_sym sigma => patt_sym sigma
    | patt_app phi1 phi2 => patt_app (bevar_subst phi1 psi x)
                                     (bevar_subst phi2 psi x)
    | patt_bott => patt_bott
    | patt_imp phi1 phi2 => patt_imp (bevar_subst phi1 psi x) (bevar_subst phi2 psi x)
    | patt_exists phi' => patt_exists (bevar_subst phi' psi (S x))
    | patt_mu phi' => patt_mu (bevar_subst phi' psi x)
    end.

  Fixpoint bsvar_subst (phi psi : Pattern) (x : db_index) :=
    match phi with
    | patt_free_evar x' => patt_free_evar x'
    | patt_free_svar x' => patt_free_svar x'
    | patt_bound_evar n => patt_bound_evar n
    | patt_bound_svar n => match compare_nat n x with
                           | Nat_less _ _ _ => patt_bound_svar n
                           | Nat_equal _ _ _ => psi
                           | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
                           end
    | patt_sym sigma => patt_sym sigma
    | patt_app phi1 phi2 => patt_app (bsvar_subst phi1 psi x)
                                     (bsvar_subst phi2 psi x)
    | patt_bott => patt_bott
    | patt_imp phi1 phi2 => patt_imp (bsvar_subst phi1 psi x) (bsvar_subst phi2 psi x)
    | patt_exists phi' => patt_exists (bsvar_subst phi' psi x)
    | patt_mu phi' => patt_mu (bsvar_subst phi' psi (S x))
    end.


  Fixpoint bevar_occur (phi : Pattern) (x : db_index) : bool :=
    match phi with
    | patt_free_evar x' => false
    | patt_free_svar x' => false
    | patt_bound_evar n => if decide (n = x) is left _ then true else false
    | patt_bound_svar n => false
    | patt_sym sigma => false
    | patt_app phi1 phi2 => orb (bevar_occur phi1 x)
                                (bevar_occur phi2 x)
    | patt_bott => false
    | patt_imp phi1 phi2 => orb (bevar_occur phi1 x) (bevar_occur phi2 x)
    | patt_exists phi' => bevar_occur phi' (S x)
    | patt_mu phi' => bevar_occur phi' x
    end.

  Fixpoint bsvar_occur (phi : Pattern) (x : db_index) : bool :=
    match phi with
    | patt_free_evar x' => false
    | patt_free_svar x' => false
    | patt_bound_evar n => false
    | patt_bound_svar n => if (decide (n = x)) is left _ then true else false
    | patt_sym sigma => false
    | patt_app phi1 phi2 => orb (bsvar_occur phi1 x)
                                (bsvar_occur phi2 x)
    | patt_bott => false
    | patt_imp phi1 phi2 => orb (bsvar_occur phi1 x) (bsvar_occur phi2 x)
    | patt_exists phi' => bsvar_occur phi' x
    | patt_mu phi' => bsvar_occur phi' (S x)
    end.
  
  (* substitute free element variable x for psi in phi *)
  Fixpoint free_evar_subst (phi psi : Pattern) (x : evar) :=
    match phi with
    | patt_free_evar x' => if decide (x = x') is left _ then psi else patt_free_evar x'
    | patt_free_svar X => patt_free_svar X
    | patt_bound_evar x' => patt_bound_evar x'
    | patt_bound_svar X => patt_bound_svar X
    | patt_sym sigma => patt_sym sigma
    | patt_app phi1 phi2 => patt_app (free_evar_subst phi1 psi x)
                                     (free_evar_subst phi2 psi x)
    | patt_bott => patt_bott
    | patt_imp phi1 phi2 => patt_imp (free_evar_subst phi1 psi x) (free_evar_subst phi2 psi x)
    | patt_exists phi' => patt_exists (free_evar_subst phi' psi x)
    | patt_mu phi' => patt_mu (free_evar_subst phi' psi x)
    end.

  (* substitute free set variable X for psi in phi *)
  Fixpoint free_svar_subst (phi psi : Pattern) (X : svar) : Pattern :=
    match phi with
    | patt_free_evar x => patt_free_evar x
    | patt_free_svar X' => if decide (X = X') is left _ then psi else patt_free_svar X'
    | patt_bound_evar x => patt_bound_evar x
    | patt_bound_svar X' => patt_bound_svar X'
    | patt_sym sigma => patt_sym sigma
    | patt_app phi1 phi2 => patt_app (free_svar_subst phi1 psi X)
                                     (free_svar_subst phi2 psi X)
    | patt_bott => patt_bott
    | patt_imp phi1 phi2 => patt_imp (free_svar_subst phi1 psi X) (free_svar_subst phi2 psi X)
    | patt_exists phi' => patt_exists (free_svar_subst phi' psi X)
    | patt_mu phi' => patt_mu (free_svar_subst phi' psi X)
    end.

  (* instantiate exists x. p or mu x. p with psi for p *)
  Definition instantiate (phi psi : Pattern) :=
    match phi with
    | patt_exists phi' => bevar_subst phi' psi 0
    | patt_mu phi' => bsvar_subst phi' psi 0
    | _ => phi
    end.

  (* replace element variable x with de Bruijn index level *)
  Fixpoint evar_quantify (x : evar) (level : db_index)
           (p : Pattern) : Pattern :=
    match p with
    | patt_free_evar x' => if decide (x = x') is left _ then patt_bound_evar level else patt_free_evar x'
    | patt_free_svar x' => patt_free_svar x'
    | patt_bound_evar x' => patt_bound_evar x'
    | patt_bound_svar X => patt_bound_svar X
    | patt_sym s => patt_sym s
    | patt_app ls rs => patt_app (evar_quantify x level ls) (evar_quantify x level rs)
    | patt_bott => patt_bott
    | patt_imp ls rs => patt_imp (evar_quantify x level ls) (evar_quantify x level rs)
    | patt_exists p' => patt_exists (evar_quantify x (S level) p')
    | patt_mu p' => patt_mu (evar_quantify x level p')
    end.

  (* replace set variable X with de Bruijn index level *)
  Fixpoint svar_quantify (X : svar) (level : db_index)
           (p : Pattern) : Pattern :=
    match p with
    | patt_free_evar x' => patt_free_evar x'
    | patt_free_svar X' => if decide (X = X') is left _ then patt_bound_svar level else patt_free_svar X'
    | patt_bound_evar x' => patt_bound_evar x'
    | patt_bound_svar X => patt_bound_svar X
    | patt_sym s => patt_sym s
    | patt_app ls rs => patt_app (svar_quantify X level ls) (svar_quantify X level rs)
    | patt_bott => patt_bott
    | patt_imp ls rs => patt_imp (svar_quantify X level ls) (svar_quantify X level rs)
    | patt_exists p' => patt_exists (svar_quantify X level p')
    | patt_mu p' => patt_mu (svar_quantify X (S level) p')
    end.

  Definition exists_quantify (x : evar)
             (p : Pattern) : Pattern :=
    patt_exists (evar_quantify x 0 p).

  Definition mu_quantify (X : svar)
             (p : Pattern) : Pattern :=
    patt_mu (svar_quantify X 0 p).


  
  (* replace de Bruijn index k with element variable n *)
  Definition evar_open (k : db_index) (x : evar) (p : Pattern) : Pattern :=
    bevar_subst p (patt_free_evar x) k.


  (* replace de Bruijn index k with set variable n *)
  Definition svar_open (k : db_index) (X : svar) (p : Pattern) : Pattern :=
    bsvar_subst p (patt_free_svar X) k.

  Lemma evar_open_free_evar k n x: evar_open k n (patt_free_evar x) = patt_free_evar x.
Σ: Signature
k: db_index
n, x: evar
evar_open k n (patt_free_evar x) = patt_free_evar x
  Proof.
Σ: Signature
k: db_index
n, x: evar
evar_open k n (patt_free_evar x) = patt_free_evar x reflexivity. Qed.
  Lemma evar_open_free_svar k n X: evar_open k n (patt_free_svar X) = patt_free_svar X.
Σ: Signature
k: db_index
n: evar
X: svar
evar_open k n (patt_free_svar X) = patt_free_svar X
  Proof.
Σ: Signature
k: db_index
n: evar
X: svar
evar_open k n (patt_free_svar X) = patt_free_svar X reflexivity. Qed.
  Lemma evar_open_bound_evar k n x: evar_open k n (patt_bound_evar x) = 
                           match compare_nat x k with
                           | Nat_less _ _ _ => patt_bound_evar x
                           | Nat_equal _ _ _ => patt_free_evar n
                           | Nat_greater _ _ _ => patt_bound_evar (Nat.pred x)
                           end.
Σ: Signature
k: db_index
n: evar
x: db_index
evar_open k n (patt_bound_evar x) =
match compare_nat x k with
| Nat_less _ _ _ => patt_bound_evar x
| Nat_equal _ _ _ => patt_free_evar n
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred x)
end
  Proof.
Σ: Signature
k: db_index
n: evar
x: db_index
evar_open k n (patt_bound_evar x) =
match compare_nat x k with
| Nat_less _ _ _ => patt_bound_evar x
| Nat_equal _ _ _ => patt_free_evar n
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred x)
end
    cbn.
Σ: Signature
k: db_index
n: evar
x: db_index
match compare_nat x k with
| Nat_less _ _ _ => patt_bound_evar x
| Nat_equal _ _ _ => patt_free_evar n
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred x)
end =
match compare_nat x k with
| Nat_less _ _ _ => patt_bound_evar x
| Nat_equal _ _ _ => patt_free_evar n
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred x)
end case_match; done.
  Qed.
  Lemma evar_open_bound_svar k n X: evar_open k n (patt_bound_svar X) = patt_bound_svar X.
Σ: Signature
k: db_index
n: evar
X: db_index
evar_open k n (patt_bound_svar X) = patt_bound_svar X
  Proof.
Σ: Signature
k: db_index
n: evar
X: db_index
evar_open k n (patt_bound_svar X) = patt_bound_svar X reflexivity. Qed.
  Lemma evar_open_sym k n s: evar_open k n (patt_sym s) = patt_sym s.
Σ: Signature
k: db_index
n: evar
s: symbols
evar_open k n (patt_sym s) = patt_sym s
  Proof.
Σ: Signature
k: db_index
n: evar
s: symbols
evar_open k n (patt_sym s) = patt_sym s reflexivity. Qed.
  Lemma evar_open_app k n ls rs: evar_open k n (patt_app ls rs) = patt_app (evar_open k n ls) (evar_open k n rs).
Σ: Signature
k: db_index
n: evar
ls, rs: Pattern
evar_open k n (patt_app ls rs) =
patt_app (evar_open k n ls) (evar_open k n rs)
  Proof.
Σ: Signature
k: db_index
n: evar
ls, rs: Pattern
evar_open k n (patt_app ls rs) =
patt_app (evar_open k n ls) (evar_open k n rs) reflexivity. Qed.
  Lemma evar_open_bott k n: evar_open k n patt_bott = patt_bott.
Σ: Signature
k: db_index
n: evar
evar_open k n patt_bott = patt_bott
  Proof.
Σ: Signature
k: db_index
n: evar
evar_open k n patt_bott = patt_bott reflexivity. Qed.
  Lemma evar_open_imp k n ls rs: evar_open k n (patt_imp ls rs) = patt_imp (evar_open k n ls) (evar_open k n rs).
Σ: Signature
k: db_index
n: evar
ls, rs: Pattern
evar_open k n (patt_imp ls rs) =
patt_imp (evar_open k n ls) (evar_open k n rs)
  Proof.
Σ: Signature
k: db_index
n: evar
ls, rs: Pattern
evar_open k n (patt_imp ls rs) =
patt_imp (evar_open k n ls) (evar_open k n rs) reflexivity. Qed.
  Lemma evar_open_exists k n p': evar_open k n (patt_exists p') = patt_exists (evar_open (S k) n p').
Σ: Signature
k: db_index
n: evar
p': Pattern
evar_open k n (patt_exists p') =
patt_exists (evar_open (S k) n p')
  Proof.
Σ: Signature
k: db_index
n: evar
p': Pattern
evar_open k n (patt_exists p') =
patt_exists (evar_open (S k) n p') reflexivity. Qed.
  Lemma evar_open_mu k n p': evar_open k n (patt_mu p') = patt_mu (evar_open k n p').
Σ: Signature
k: db_index
n: evar
p': Pattern
evar_open k n (patt_mu p') =
patt_mu (evar_open k n p')
  Proof.
Σ: Signature
k: db_index
n: evar
p': Pattern
evar_open k n (patt_mu p') =
patt_mu (evar_open k n p') reflexivity. Qed.

  (* More trivial but useful lemmas *)
  Lemma svar_open_free_evar k n x: svar_open k n (patt_free_evar x) = patt_free_evar x.
Σ: Signature
k: db_index
n: svar
x: evar
svar_open k n (patt_free_evar x) = patt_free_evar x
  Proof.
Σ: Signature
k: db_index
n: svar
x: evar
svar_open k n (patt_free_evar x) = patt_free_evar x reflexivity. Qed.
  Lemma svar_open_free_svar k n X: svar_open k n (patt_free_svar X) = patt_free_svar X.
Σ: Signature
k: db_index
n, X: svar
svar_open k n (patt_free_svar X) = patt_free_svar X
  Proof.
Σ: Signature
k: db_index
n, X: svar
svar_open k n (patt_free_svar X) = patt_free_svar X reflexivity. Qed.
  Lemma svar_open_bound_evar k n x: svar_open k n (patt_bound_evar x) = patt_bound_evar x.
Σ: Signature
k: db_index
n: svar
x: db_index
svar_open k n (patt_bound_evar x) = patt_bound_evar x
  Proof.
Σ: Signature
k: db_index
n: svar
x: db_index
svar_open k n (patt_bound_evar x) = patt_bound_evar x reflexivity. Qed.
  Lemma svar_open_bound_svar k n X: svar_open k n (patt_bound_svar X) = 
                                    match compare_nat X k with
                                    | Nat_less _ _ _ => patt_bound_svar X
                                    | Nat_equal _ _ _ => patt_free_svar n
                                    | Nat_greater _ _ _ => patt_bound_svar (Nat.pred X)
                                    end.
Σ: Signature
k: db_index
n: svar
X: db_index
svar_open k n (patt_bound_svar X) =
match compare_nat X k with
| Nat_less _ _ _ => patt_bound_svar X
| Nat_equal _ _ _ => patt_free_svar n
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred X)
end
  Proof.
Σ: Signature
k: db_index
n: svar
X: db_index
svar_open k n (patt_bound_svar X) =
match compare_nat X k with
| Nat_less _ _ _ => patt_bound_svar X
| Nat_equal _ _ _ => patt_free_svar n
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred X)
end
    reflexivity.
  Qed.
  Lemma svar_open_sym k n s: svar_open k n (patt_sym s) = patt_sym s.
Σ: Signature
k: db_index
n: svar
s: symbols
svar_open k n (patt_sym s) = patt_sym s
  Proof.
Σ: Signature
k: db_index
n: svar
s: symbols
svar_open k n (patt_sym s) = patt_sym s reflexivity. Qed.
  Lemma svar_open_app k n ls rs: svar_open k n (patt_app ls rs) = patt_app (svar_open k n ls) (svar_open k n rs).
Σ: Signature
k: db_index
n: svar
ls, rs: Pattern
svar_open k n (patt_app ls rs) =
patt_app (svar_open k n ls) (svar_open k n rs)
  Proof.
Σ: Signature
k: db_index
n: svar
ls, rs: Pattern
svar_open k n (patt_app ls rs) =
patt_app (svar_open k n ls) (svar_open k n rs) reflexivity. Qed.
  Lemma svar_open_bott k n: svar_open k n patt_bott = patt_bott.
Σ: Signature
k: db_index
n: svar
svar_open k n patt_bott = patt_bott
  Proof.
Σ: Signature
k: db_index
n: svar
svar_open k n patt_bott = patt_bott reflexivity. Qed.
  Lemma svar_open_imp k n ls rs: svar_open k n (patt_imp ls rs) = patt_imp (svar_open k n ls) (svar_open k n rs).
Σ: Signature
k: db_index
n: svar
ls, rs: Pattern
svar_open k n (patt_imp ls rs) =
patt_imp (svar_open k n ls) (svar_open k n rs)
  Proof.
Σ: Signature
k: db_index
n: svar
ls, rs: Pattern
svar_open k n (patt_imp ls rs) =
patt_imp (svar_open k n ls) (svar_open k n rs) reflexivity. Qed.
  Lemma svar_open_exists k n p': svar_open k n (patt_exists p') = patt_exists (svar_open k n p').
Σ: Signature
k: db_index
n: svar
p': Pattern
svar_open k n (patt_exists p') =
patt_exists (svar_open k n p')
  Proof.
Σ: Signature
k: db_index
n: svar
p': Pattern
svar_open k n (patt_exists p') =
patt_exists (svar_open k n p') reflexivity. Qed.
  Lemma svar_open_mu k n p': svar_open k n (patt_mu p') = patt_mu (svar_open (S k) n p').
Σ: Signature
k: db_index
n: svar
p': Pattern
svar_open k n (patt_mu p') =
patt_mu (svar_open (S k) n p')
  Proof.
Σ: Signature
k: db_index
n: svar
p': Pattern
svar_open k n (patt_mu p') =
patt_mu (svar_open (S k) n p') reflexivity. Qed.


  Lemma evar_open_size :
    forall (k : db_index) (n : evar) (p : Pattern),
      size p = size (evar_open k n p).
Σ: Signature
∀ (k : db_index) (n : evar) (p : Pattern),
  size p = size (evar_open k n p)
  Proof.
Σ: Signature
∀ (k : db_index) (n : evar) (p : Pattern),
  size p = size (evar_open k n p)
    intros k n p.
Σ: Signature
k: db_index
n: evar
p: Pattern
size p = size (evar_open k n p) generalize dependent k.
Σ: Signature
n: evar
p: Pattern
∀ k : db_index, size p = size (evar_open k n p)
    induction p; intros k; cbn; try reflexivity.
Σ: Signature
n: evar
n0, k: db_index
0 =
size
  match compare_nat n0 k with
  | Nat_less _ _ _ => patt_bound_evar n0
  | Nat_equal _ _ _ => patt_free_evar n
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n0)
  end
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size p1 = size (evar_open k n p1)
IHp2: ∀ k : db_index,
  size p2 = size (evar_open k n p2)
k: db_index
S (size p1 + size p2) =
S
  (size (bevar_subst p1 (patt_free_evar n) k) +
   size (bevar_subst p2 (patt_free_evar n) k))
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size p1 = size (evar_open k n p1)
IHp2: ∀ k : db_index,
  size p2 = size (evar_open k n p2)
k: db_index
S (size p1 + size p2) =
S
  (size (bevar_subst p1 (patt_free_evar n) k) +
   size (bevar_subst p2 (patt_free_evar n) k))
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size (evar_open k n p)
k: db_index
S (size p) =
S (size (bevar_subst p (patt_free_evar n) (S k)))
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size (evar_open k n p)
k: db_index
S (size p) =
S (size (bevar_subst p (patt_free_evar n) k))
    break_match_goal; reflexivity.
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size p1 = size (evar_open k n p1)
IHp2: ∀ k : db_index,
  size p2 = size (evar_open k n p2)
k: db_index
S (size p1 + size p2) =
S
  (size (bevar_subst p1 (patt_free_evar n) k) +
   size (bevar_subst p2 (patt_free_evar n) k))
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size p1 = size (evar_open k n p1)
IHp2: ∀ k : db_index,
  size p2 = size (evar_open k n p2)
k: db_index
S (size p1 + size p2) =
S
  (size (bevar_subst p1 (patt_free_evar n) k) +
   size (bevar_subst p2 (patt_free_evar n) k))
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size (evar_open k n p)
k: db_index
S (size p) =
S (size (bevar_subst p (patt_free_evar n) (S k)))
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size (evar_open k n p)
k: db_index
S (size p) =
S (size (bevar_subst p (patt_free_evar n) k))
    rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size p1 = size (evar_open k n p1)
IHp2: ∀ k : db_index,
  size p2 = size (evar_open k n p2)
k: db_index
S (size p1 + size p2) =
S
  (size (bevar_subst p1 (patt_free_evar n) k) +
   size (bevar_subst p2 (patt_free_evar n) k))
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size (evar_open k n p)
k: db_index
S (size p) =
S (size (bevar_subst p (patt_free_evar n) (S k)))
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size (evar_open k n p)
k: db_index
S (size p) =
S (size (bevar_subst p (patt_free_evar n) k))
    rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size (evar_open k n p)
k: db_index
S (size p) =
S (size (bevar_subst p (patt_free_evar n) (S k)))
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size (evar_open k n p)
k: db_index
S (size p) =
S (size (bevar_subst p (patt_free_evar n) k))
    rewrite (IHp (S k)); reflexivity.
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index, size p = size (evar_open k n p)
k: db_index
S (size p) =
S (size (bevar_subst p (patt_free_evar n) k))
    rewrite (IHp k); reflexivity.
  Qed.

  Lemma svar_open_size :
    forall (k : db_index) (n : svar) (p : Pattern),
      size p = size (svar_open k n p).
Σ: Signature
∀ (k : db_index) (n : svar) (p : Pattern),
  size p = size (svar_open k n p)
  Proof.
Σ: Signature
∀ (k : db_index) (n : svar) (p : Pattern),
  size p = size (svar_open k n p)
    intros k n p.
Σ: Signature
k: db_index
n: svar
p: Pattern
size p = size (svar_open k n p) generalize dependent k.
Σ: Signature
n: svar
p: Pattern
∀ k : db_index, size p = size (svar_open k n p)
    induction p; intros k; cbn; try reflexivity.
Σ: Signature
n: svar
n0, k: db_index
0 =
size
  match compare_nat n0 k with
  | Nat_less _ _ _ => patt_bound_svar n0
  | Nat_equal _ _ _ => patt_free_svar n
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n0)
  end
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size p1 = size (svar_open k n p1)
IHp2: ∀ k : db_index,
  size p2 = size (svar_open k n p2)
k: db_index
S (size p1 + size p2) =
S
  (size (bsvar_subst p1 (patt_free_svar n) k) +
   size (bsvar_subst p2 (patt_free_svar n) k))
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size p1 = size (svar_open k n p1)
IHp2: ∀ k : db_index,
  size p2 = size (svar_open k n p2)
k: db_index
S (size p1 + size p2) =
S
  (size (bsvar_subst p1 (patt_free_svar n) k) +
   size (bsvar_subst p2 (patt_free_svar n) k))
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size (svar_open k n p)
k: db_index
S (size p) =
S (size (bsvar_subst p (patt_free_svar n) k))
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size (svar_open k n p)
k: db_index
S (size p) =
S (size (bsvar_subst p (patt_free_svar n) (S k)))
    break_match_goal; reflexivity.
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size p1 = size (svar_open k n p1)
IHp2: ∀ k : db_index,
  size p2 = size (svar_open k n p2)
k: db_index
S (size p1 + size p2) =
S
  (size (bsvar_subst p1 (patt_free_svar n) k) +
   size (bsvar_subst p2 (patt_free_svar n) k))
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size p1 = size (svar_open k n p1)
IHp2: ∀ k : db_index,
  size p2 = size (svar_open k n p2)
k: db_index
S (size p1 + size p2) =
S
  (size (bsvar_subst p1 (patt_free_svar n) k) +
   size (bsvar_subst p2 (patt_free_svar n) k))
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size (svar_open k n p)
k: db_index
S (size p) =
S (size (bsvar_subst p (patt_free_svar n) k))
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size (svar_open k n p)
k: db_index
S (size p) =
S (size (bsvar_subst p (patt_free_svar n) (S k)))
    rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size p1 = size (svar_open k n p1)
IHp2: ∀ k : db_index,
  size p2 = size (svar_open k n p2)
k: db_index
S (size p1 + size p2) =
S
  (size (bsvar_subst p1 (patt_free_svar n) k) +
   size (bsvar_subst p2 (patt_free_svar n) k))
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size (svar_open k n p)
k: db_index
S (size p) =
S (size (bsvar_subst p (patt_free_svar n) k))
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size (svar_open k n p)
k: db_index
S (size p) =
S (size (bsvar_subst p (patt_free_svar n) (S k)))
    rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size (svar_open k n p)
k: db_index
S (size p) =
S (size (bsvar_subst p (patt_free_svar n) k))
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size (svar_open k n p)
k: db_index
S (size p) =
S (size (bsvar_subst p (patt_free_svar n) (S k)))
    rewrite (IHp k); reflexivity.
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index, size p = size (svar_open k n p)
k: db_index
S (size p) =
S (size (bsvar_subst p (patt_free_svar n) (S k)))
    rewrite (IHp (S k)); reflexivity.
  Qed.


 
   (* From https://www.chargueraud.org/research/2009/ln/main.pdf in 3.3 (body def.) *)
   Definition wfc_body_ex phi  := forall x, 
   ~ elem_of x (free_evars phi) -> well_formed_closed (evar_open 0 x phi) = true.


   Lemma positive_negative_occurrence_evar_open_and : forall (phi : Pattern) (db1 db2 : db_index) (x : evar),
   (*le db1 db2 ->*)
   (no_positive_occurrence_db_b db1 phi -> no_positive_occurrence_db_b db1 (evar_open db2 x phi))
   /\ (no_negative_occurrence_db_b db1 phi -> no_negative_occurrence_db_b db1 (evar_open db2 x phi)).
Σ: Signature
∀ (phi : Pattern) (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi))
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi))
Proof.
Σ: Signature
∀ (phi : Pattern) (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi))
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi))
 induction phi; intros db1 db2 x'; cbn; split; intro H; try lia; auto.
Σ: Signature
n, db1, db2: db_index
x': evar
H: true
no_positive_occurrence_db_b db1
  match compare_nat n db2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x'
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end
Σ: Signature
n, db1, db2: db_index
x': evar
H: true
no_negative_occurrence_db_b db1
  match compare_nat n db2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x'
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi1))
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi1))
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi2))
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi2))
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1 phi1 &&
no_positive_occurrence_db_b db1 phi2
no_positive_occurrence_db_b db1
  (bevar_subst phi1 (patt_free_evar x') db2) &&
no_positive_occurrence_db_b db1
  (bevar_subst phi2 (patt_free_evar x') db2)
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi1))
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi1))
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi2))
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi2))
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1 phi1 &&
no_negative_occurrence_db_b db1 phi2
no_negative_occurrence_db_b db1
  (bevar_subst phi1 (patt_free_evar x') db2) &&
no_negative_occurrence_db_b db1
  (bevar_subst phi2 (patt_free_evar x') db2)
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi1))
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi1))
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi2))
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi2))
db1, db2: db_index
x': evar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1 phi1 &&
no_positive_occurrence_db_b db1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  (bevar_subst phi1 (patt_free_evar x') db2) &&
no_positive_occurrence_db_b db1
  (bevar_subst phi2 (patt_free_evar x') db2)
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi1))
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi1))
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi2))
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi2))
db1, db2: db_index
x': evar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1 phi1 &&
no_negative_occurrence_db_b db1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  (bevar_subst phi1 (patt_free_evar x') db2) &&
no_negative_occurrence_db_b db1
  (bevar_subst phi2 (patt_free_evar x') db2)
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi))
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi))
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1 phi
no_positive_occurrence_db_b db1
  (bevar_subst phi (patt_free_evar x') (S db2))
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi))
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi))
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1 phi
no_negative_occurrence_db_b db1
  (bevar_subst phi (patt_free_evar x') (S db2))
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi))
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi))
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b (S db1) phi
no_positive_occurrence_db_b 
  (S db1) (bevar_subst phi (patt_free_evar x') db2)
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi))
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi))
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b (S db1) phi
no_negative_occurrence_db_b 
  (S db1) (bevar_subst phi (patt_free_evar x') db2)
 *
Σ: Signature
n, db1, db2: db_index
x': evar
H: true
no_positive_occurrence_db_b db1
  match compare_nat n db2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x'
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end case_match; auto.
 *
Σ: Signature
n, db1, db2: db_index
x': evar
H: true
no_negative_occurrence_db_b db1
  match compare_nat n db2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x'
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end case_match; auto.
 *
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi1))
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi1))
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi2))
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi2))
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1 phi1 &&
no_positive_occurrence_db_b db1 phi2
no_positive_occurrence_db_b db1
  (bevar_subst phi1 (patt_free_evar x') db2) &&
no_positive_occurrence_db_b db1
  (bevar_subst phi2 (patt_free_evar x') db2) rewrite -> (proj1 (IHphi1 db1 db2 x')), -> (proj1 (IHphi2 db1 db2 x')); destruct_and!; auto.
 *
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi1))
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi1))
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi2))
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi2))
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1 phi1 &&
no_negative_occurrence_db_b db1 phi2
no_negative_occurrence_db_b db1
  (bevar_subst phi1 (patt_free_evar x') db2) &&
no_negative_occurrence_db_b db1
  (bevar_subst phi2 (patt_free_evar x') db2) rewrite -> (proj2 (IHphi1 db1 db2 x')), -> (proj2 (IHphi2 db1 db2 x')); destruct_and!; auto.
 *
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi1))
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi1))
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi2))
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi2))
db1, db2: db_index
x': evar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1 phi1 &&
no_positive_occurrence_db_b db1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  (bevar_subst phi1 (patt_free_evar x') db2) &&
no_positive_occurrence_db_b db1
  (bevar_subst phi2 (patt_free_evar x') db2) rewrite -> (proj2 (IHphi1 db1 db2 x')), -> (proj1 (IHphi2 db1 db2 x')); destruct_and!; auto.
 *
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi1
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi1))
  ∧ (no_negative_occurrence_db_b db1 phi1
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi1))
IHphi2: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi2
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi2))
  ∧ (no_negative_occurrence_db_b db1 phi2
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi2))
db1, db2: db_index
x': evar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1 phi1 &&
no_negative_occurrence_db_b db1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  (bevar_subst phi1 (patt_free_evar x') db2) &&
no_negative_occurrence_db_b db1
  (bevar_subst phi2 (patt_free_evar x') db2) rewrite -> (proj1 (IHphi1 db1 db2 x')), -> (proj2 (IHphi2 db1 db2 x')); destruct_and!; auto.
 *
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi))
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi))
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1 phi
no_positive_occurrence_db_b db1
  (bevar_subst phi (patt_free_evar x') (S db2)) rewrite -> (proj1 (IHphi db1 (S db2) x')); auto.
 *
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi))
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi))
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1 phi
no_negative_occurrence_db_b db1
  (bevar_subst phi (patt_free_evar x') (S db2)) rewrite -> (proj2 (IHphi db1 (S db2) x')); auto.
 *
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi))
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi))
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b (S db1) phi
no_positive_occurrence_db_b (S db1)
  (bevar_subst phi (patt_free_evar x') db2) rewrite -> (proj1 (IHphi (S db1) db2 x')); auto.
 *
Σ: Signature
phi: Pattern
IHphi: ∀ (db1 db2 : db_index) (x : evar),
  (no_positive_occurrence_db_b db1 phi
   → no_positive_occurrence_db_b db1
       (evar_open db2 x phi))
  ∧ (no_negative_occurrence_db_b db1 phi
     → no_negative_occurrence_db_b db1
         (evar_open db2 x phi))
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b (S db1) phi
no_negative_occurrence_db_b (S db1)
  (bevar_subst phi (patt_free_evar x') db2) rewrite -> (proj2 (IHphi (S db1) db2 x')); auto.
Qed.

Lemma no_negative_occurrence_evar_open phi db1 db2 x:
 no_negative_occurrence_db_b db1 phi = true ->
 no_negative_occurrence_db_b db1 (evar_open db2 x phi) = true.
Σ: Signature
phi: Pattern
db1, db2: db_index
x: evar
no_negative_occurrence_db_b db1 phi = true
→ no_negative_occurrence_db_b db1
    (evar_open db2 x phi) = true
Proof.
Σ: Signature
phi: Pattern
db1, db2: db_index
x: evar
no_negative_occurrence_db_b db1 phi = true
→ no_negative_occurrence_db_b db1
    (evar_open db2 x phi) = true
 apply positive_negative_occurrence_evar_open_and.
Qed.

Lemma no_positive_occurrence_evar_open phi db1 db2 x:
 no_positive_occurrence_db_b db1 phi = true ->
 no_positive_occurrence_db_b db1 (evar_open db2 x phi) = true.
Σ: Signature
phi: Pattern
db1, db2: db_index
x: evar
no_positive_occurrence_db_b db1 phi = true
→ no_positive_occurrence_db_b db1
    (evar_open db2 x phi) = true
Proof.
Σ: Signature
phi: Pattern
db1, db2: db_index
x: evar
no_positive_occurrence_db_b db1 phi = true
→ no_positive_occurrence_db_b db1
    (evar_open db2 x phi) = true
 apply positive_negative_occurrence_evar_open_and.
Qed.


(*Helper lemma for wf_body_to_wf_ex*)
Lemma wfc_ex_aux_body_ex_imp2:
 forall phi n x,
   well_formed_closed_ex_aux (evar_open n x phi) n = true
   ->
   well_formed_closed_ex_aux phi (S n) = true.
Σ: Signature
∀ (phi : Pattern) (n : db_index) (x : evar),
  well_formed_closed_ex_aux (evar_open n x phi) n =
  true → well_formed_closed_ex_aux phi (S n) = true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n : db_index) (x : evar),
  well_formed_closed_ex_aux (evar_open n x phi) n =
  true → well_formed_closed_ex_aux phi (S n) = true
 induction phi; firstorder.
Σ: Signature
n, n0: db_index
x: evar
H: well_formed_closed_ex_aux
  (evar_open n0 x (patt_bound_evar n)) n0 = true
well_formed_closed_ex_aux (patt_bound_evar n) (S n0) =
true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (evar_open n x (patt_app phi1 phi2)) n = true
well_formed_closed_ex_aux (patt_app phi1 phi2) (S n) =
true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (evar_open n x (patt_imp phi1 phi2)) n = true
well_formed_closed_ex_aux (patt_imp phi1 phi2) (S n) =
true
 -
Σ: Signature
n, n0: db_index
x: evar
H: well_formed_closed_ex_aux
  (evar_open n0 x (patt_bound_evar n)) n0 = true
well_formed_closed_ex_aux (patt_bound_evar n) (S n0) =
true simpl.
Σ: Signature
n, n0: db_index
x: evar
H: well_formed_closed_ex_aux
  (evar_open n0 x (patt_bound_evar n)) n0 = true
(if decide (n < S n0) then true else false) = true cbn in H.
Σ: Signature
n, n0: db_index
x: evar
H: well_formed_closed_ex_aux
  match compare_nat n n0 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end n0 = true
(if decide (n < S n0) then true else false) = true unfold well_formed_closed_ex_aux.
Σ: Signature
n, n0: db_index
x: evar
H: well_formed_closed_ex_aux
  match compare_nat n n0 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end n0 = true
(if decide (n < S n0) then true else false) = true
   repeat case_match; simpl; auto; try lia.
Σ: Signature
n, n0: db_index
x: evar
g: n > n0
H0: compare_nat n n0 = Nat_greater n n0 g
H: well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) n0 = true
n1: ¬ n < S n0
H1: decide (n < S n0) = right n1
false = true
   unfold well_formed_closed_ex_aux in H.
Σ: Signature
n, n0: db_index
x: evar
g: n > n0
H0: compare_nat n n0 = Nat_greater n n0 g
H: (if decide (Nat.pred n < n0) then true else false) =
true
n1: ¬ n < S n0
H1: decide (n < S n0) = right n1
false = true case_match; auto.
Σ: Signature
n, n0: db_index
x: evar
g: n > n0
H0: compare_nat n n0 = Nat_greater n n0 g
l: Nat.pred n < n0
H2: decide (Nat.pred n < n0) = left l
H: true = true
n1: ¬ n < S n0
H1: decide (n < S n0) = right n1
false = true lia.
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (evar_open n x (patt_app phi1 phi2)) n = true
well_formed_closed_ex_aux (patt_app phi1 phi2) (S n) =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n &&
well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n = true
well_formed_closed_ex_aux (patt_app phi1 phi2) (S n) =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n &&
well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n = true
well_formed_closed_ex_aux phi1 (S n) &&
well_formed_closed_ex_aux phi2 (S n) = true
   apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n = true
∧ well_formed_closed_ex_aux
    (bevar_subst phi2 (patt_free_evar x) n) n =
  true
well_formed_closed_ex_aux phi1 (S n) &&
well_formed_closed_ex_aux phi2 (S n) = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
well_formed_closed_ex_aux phi1 (S n) &&
well_formed_closed_ex_aux phi2 (S n) = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
true && well_formed_closed_ex_aux phi2 (S n) = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
well_formed_closed_ex_aux (evar_open n ?x phi1) n =
true 2: apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
true && well_formed_closed_ex_aux phi2 (S n) = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
well_formed_closed_ex_aux (evar_open n ?x phi2) n =
true 2: apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
true && true = true
   reflexivity.
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (evar_open n x (patt_imp phi1 phi2)) n = true
well_formed_closed_ex_aux (patt_imp phi1 phi2) (S n) =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n &&
well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n = true
well_formed_closed_ex_aux (patt_imp phi1 phi2) (S n) =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n &&
well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n = true
well_formed_closed_ex_aux phi1 (S n) &&
well_formed_closed_ex_aux phi2 (S n) = true
   apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n = true
∧ well_formed_closed_ex_aux
    (bevar_subst phi2 (patt_free_evar x) n) n =
  true
well_formed_closed_ex_aux phi1 (S n) &&
well_formed_closed_ex_aux phi2 (S n) = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
well_formed_closed_ex_aux phi1 (S n) &&
well_formed_closed_ex_aux phi2 (S n) = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
true && well_formed_closed_ex_aux phi2 (S n) = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
well_formed_closed_ex_aux (evar_open n ?x phi1) n =
true 2: apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
true && well_formed_closed_ex_aux phi2 (S n) = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
well_formed_closed_ex_aux (evar_open n ?x phi2) n =
true 2: apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi1) n = true
  → well_formed_closed_ex_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (x : evar),
  well_formed_closed_ex_aux
    (evar_open n x phi2) n = true
  → well_formed_closed_ex_aux phi2 (S n) =
    true
n: db_index
x: evar
H1: well_formed_closed_ex_aux
  (bevar_subst phi1 (patt_free_evar x) n) n =
true
H2: well_formed_closed_ex_aux
  (bevar_subst phi2 (patt_free_evar x) n) n =
true
true && true = true
   reflexivity.
Qed.

(*Helper lemma for wf_body_to_wf_ex*)
Lemma wfc_mu_aux_body_ex_imp2:
 forall phi n n' x,
   well_formed_closed_mu_aux (evar_open n x phi) n' = true
   ->
   well_formed_closed_mu_aux phi n' = true.
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux (evar_open n x phi) n' =
  true → well_formed_closed_mu_aux phi n' = true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux (evar_open n x phi) n' =
  true → well_formed_closed_mu_aux phi n' = true
 induction phi; firstorder.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (evar_open n x (patt_app phi1 phi2)) n' = true
well_formed_closed_mu_aux (patt_app phi1 phi2) n' =
true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (evar_open n x (patt_imp phi1 phi2)) n' = true
well_formed_closed_mu_aux (patt_imp phi1 phi2) n' =
true
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (evar_open n x (patt_app phi1 phi2)) n' = true
well_formed_closed_mu_aux (patt_app phi1 phi2) n' =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' &&
well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
well_formed_closed_mu_aux (patt_app phi1 phi2) n' =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' &&
well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
   apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
∧ well_formed_closed_mu_aux
    (bevar_subst phi2 (patt_free_evar x) n) n' =
  true
well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
true && well_formed_closed_mu_aux phi2 n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
well_formed_closed_mu_aux (evar_open ?n ?x phi1) n' =
true 2: apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
true && well_formed_closed_mu_aux phi2 n' = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
well_formed_closed_mu_aux (evar_open ?n ?x phi2) n' =
true 2: apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
true && true = true
   reflexivity.
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (evar_open n x (patt_imp phi1 phi2)) n' = true
well_formed_closed_mu_aux (patt_imp phi1 phi2) n' =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' &&
well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
well_formed_closed_mu_aux (patt_imp phi1 phi2) n' =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' &&
well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
   apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
∧ well_formed_closed_mu_aux
    (bevar_subst phi2 (patt_free_evar x) n) n' =
  true
well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
true && well_formed_closed_mu_aux phi2 n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
well_formed_closed_mu_aux (evar_open ?n ?x phi1) n' =
true 2: apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
true && well_formed_closed_mu_aux phi2 n' = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
well_formed_closed_mu_aux (evar_open ?n ?x phi2) n' =
true 2: apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi1) n' = true
  → well_formed_closed_mu_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux
    (evar_open n x phi2) n' = true
  → well_formed_closed_mu_aux phi2 n' = true
n, n': db_index
x: evar
H1: well_formed_closed_mu_aux
  (bevar_subst phi1 (patt_free_evar x) n) n' =
true
H2: well_formed_closed_mu_aux
  (bevar_subst phi2 (patt_free_evar x) n) n' =
true
true && true = true
   reflexivity.
Qed.

Lemma wfc_ex_aux_body_mu_imp2:
 forall phi n n' X,
   well_formed_closed_ex_aux (svar_open n X phi) n' = true
   ->
   well_formed_closed_ex_aux phi n' = true.
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux (svar_open n X phi) n' =
  true → well_formed_closed_ex_aux phi n' = true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux (svar_open n X phi) n' =
  true → well_formed_closed_ex_aux phi n' = true
 induction phi; firstorder.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  (svar_open n X (patt_app phi1 phi2)) n' = true
well_formed_closed_ex_aux (patt_app phi1 phi2) n' =
true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  (svar_open n X (patt_imp phi1 phi2)) n' = true
well_formed_closed_ex_aux (patt_imp phi1 phi2) n' =
true
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  (svar_open n X (patt_app phi1 phi2)) n' = true
well_formed_closed_ex_aux (patt_app phi1 phi2) n' =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' &&
well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
well_formed_closed_ex_aux (patt_app phi1 phi2) n' =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' &&
well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n' = true
   destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n' = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
true && well_formed_closed_ex_aux phi2 n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
well_formed_closed_ex_aux (svar_open ?n ?X phi1) n' =
true 2: eassumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
true && well_formed_closed_ex_aux phi2 n' = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
well_formed_closed_ex_aux (svar_open ?n ?X phi2) n' =
true 2: eassumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
true && true = true
   reflexivity.
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  (svar_open n X (patt_imp phi1 phi2)) n' = true
well_formed_closed_ex_aux (patt_imp phi1 phi2) n' =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' &&
well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
well_formed_closed_ex_aux (patt_imp phi1 phi2) n' =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' &&
well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n' = true
   destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n' = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
true && well_formed_closed_ex_aux phi2 n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
well_formed_closed_ex_aux (svar_open ?n ?X phi1) n' =
true 2: eassumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
true && well_formed_closed_ex_aux phi2 n' = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
well_formed_closed_ex_aux (svar_open ?n ?X phi2) n' =
true 2: eassumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi1) n' = true
  → well_formed_closed_ex_aux phi1 n' = true
IHphi2: ∀ (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux
    (svar_open n X phi2) n' = true
  → well_formed_closed_ex_aux phi2 n' = true
n, n': db_index
X: svar
H0: well_formed_closed_ex_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n' =
true
H1: well_formed_closed_ex_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n' =
true
true && true = true
   reflexivity.
Qed.

Lemma wfc_mu_aux_body_mu_imp2:
 forall phi n X,
   well_formed_closed_mu_aux (svar_open n X phi) n = true
   ->
   well_formed_closed_mu_aux phi (S n) = true.
Σ: Signature
∀ (phi : Pattern) (n : db_index) (X : svar),
  well_formed_closed_mu_aux (svar_open n X phi) n =
  true → well_formed_closed_mu_aux phi (S n) = true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n : db_index) (X : svar),
  well_formed_closed_mu_aux (svar_open n X phi) n =
  true → well_formed_closed_mu_aux phi (S n) = true
 induction phi; firstorder.
Σ: Signature
n, n0: db_index
X: svar
H: well_formed_closed_mu_aux
  (svar_open n0 X (patt_bound_svar n)) n0 = true
well_formed_closed_mu_aux (patt_bound_svar n) (S n0) =
true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (svar_open n X (patt_app phi1 phi2)) n = true
well_formed_closed_mu_aux (patt_app phi1 phi2) (S n) =
true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (svar_open n X (patt_imp phi1 phi2)) n = true
well_formed_closed_mu_aux (patt_imp phi1 phi2) (S n) =
true
 -
Σ: Signature
n, n0: db_index
X: svar
H: well_formed_closed_mu_aux
  (svar_open n0 X (patt_bound_svar n)) n0 = true
well_formed_closed_mu_aux (patt_bound_svar n) (S n0) =
true simpl.
Σ: Signature
n, n0: db_index
X: svar
H: well_formed_closed_mu_aux
  (svar_open n0 X (patt_bound_svar n)) n0 = true
(if decide (n < S n0) then true else false) = true cbn in H.
Σ: Signature
n, n0: db_index
X: svar
H: well_formed_closed_mu_aux
  match compare_nat n n0 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ =>
      patt_bound_svar (Nat.pred n)
  end n0 = true
(if decide (n < S n0) then true else false) = true unfold well_formed_closed_mu_aux.
Σ: Signature
n, n0: db_index
X: svar
H: well_formed_closed_mu_aux
  match compare_nat n n0 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ =>
      patt_bound_svar (Nat.pred n)
  end n0 = true
(if decide (n < S n0) then true else false) = true
   repeat case_match; simpl; auto; try lia.
Σ: Signature
n, n0: db_index
X: svar
g: n > n0
H0: compare_nat n n0 = Nat_greater n n0 g
H: well_formed_closed_mu_aux
  (patt_bound_svar (Nat.pred n)) n0 = true
n1: ¬ n < S n0
H1: decide (n < S n0) = right n1
false = true
   unfold well_formed_closed_mu_aux in H.
Σ: Signature
n, n0: db_index
X: svar
g: n > n0
H0: compare_nat n n0 = Nat_greater n n0 g
H: (if decide (Nat.pred n < n0) then true else false) =
true
n1: ¬ n < S n0
H1: decide (n < S n0) = right n1
false = true case_match; auto.
Σ: Signature
n, n0: db_index
X: svar
g: n > n0
H0: compare_nat n n0 = Nat_greater n n0 g
l: Nat.pred n < n0
H2: decide (Nat.pred n < n0) = left l
H: true = true
n1: ¬ n < S n0
H1: decide (n < S n0) = right n1
false = true lia.
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (svar_open n X (patt_app phi1 phi2)) n = true
well_formed_closed_mu_aux (patt_app phi1 phi2) (S n) =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n &&
well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n = true
well_formed_closed_mu_aux (patt_app phi1 phi2) (S n) =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n &&
well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n = true
well_formed_closed_mu_aux phi1 (S n) &&
well_formed_closed_mu_aux phi2 (S n) = true
   apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n = true
∧ well_formed_closed_mu_aux
    (bsvar_subst phi2 (patt_free_svar X) n) n =
  true
well_formed_closed_mu_aux phi1 (S n) &&
well_formed_closed_mu_aux phi2 (S n) = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
well_formed_closed_mu_aux phi1 (S n) &&
well_formed_closed_mu_aux phi2 (S n) = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
true && well_formed_closed_mu_aux phi2 (S n) = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
well_formed_closed_mu_aux (svar_open n ?X phi1) n =
true 2: apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
true && well_formed_closed_mu_aux phi2 (S n) = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
well_formed_closed_mu_aux (svar_open n ?X phi2) n =
true 2: apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
true && true = true
   reflexivity.
 -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (svar_open n X (patt_imp phi1 phi2)) n = true
well_formed_closed_mu_aux (patt_imp phi1 phi2) (S n) =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n &&
well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n = true
well_formed_closed_mu_aux (patt_imp phi1 phi2) (S n) =
true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n &&
well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n = true
well_formed_closed_mu_aux phi1 (S n) &&
well_formed_closed_mu_aux phi2 (S n) = true
   apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n = true
∧ well_formed_closed_mu_aux
    (bsvar_subst phi2 (patt_free_svar X) n) n =
  true
well_formed_closed_mu_aux phi1 (S n) &&
well_formed_closed_mu_aux phi2 (S n) = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
well_formed_closed_mu_aux phi1 (S n) &&
well_formed_closed_mu_aux phi2 (S n) = true
   erewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
true && well_formed_closed_mu_aux phi2 (S n) = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
well_formed_closed_mu_aux (svar_open n ?X phi1) n =
true 2: apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
true && well_formed_closed_mu_aux phi2 (S n) = true
   erewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
true && true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
well_formed_closed_mu_aux (svar_open n ?X phi2) n =
true 2: apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi1) n = true
  → well_formed_closed_mu_aux phi1 (S n) =
    true
IHphi2: ∀ (n : db_index) (X : svar),
  well_formed_closed_mu_aux
    (svar_open n X phi2) n = true
  → well_formed_closed_mu_aux phi2 (S n) =
    true
n: db_index
X: svar
H1: well_formed_closed_mu_aux
  (bsvar_subst phi1 (patt_free_svar X) n) n =
true
H2: well_formed_closed_mu_aux
  (bsvar_subst phi2 (patt_free_svar X) n) n =
true
true && true = true
   reflexivity.
Qed.


  (* The following lemmas are trivial but useful for [rewrite !simpl_evar_open]. *)

  Lemma bevar_subst_free_evar ψ (pf : well_formed_closed ψ) n x :
    bevar_subst (patt_free_evar x) ψ n = patt_free_evar x.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
x: evar
bevar_subst (patt_free_evar x) ψ n = patt_free_evar x
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
x: evar
bevar_subst (patt_free_evar x) ψ n = patt_free_evar x reflexivity. Qed.

  Lemma bevar_subst_free_svar ψ (pf : well_formed_closed ψ) n X :
    bevar_subst (patt_free_svar X) ψ n = patt_free_svar X.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
X: svar
bevar_subst (patt_free_svar X) ψ n = patt_free_svar X
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
X: svar
bevar_subst (patt_free_svar X) ψ n = patt_free_svar X reflexivity. Qed.

  Lemma bevar_subst_bound_evar ψ (pf : well_formed_closed ψ) n x :
    bevar_subst (patt_bound_evar x) ψ n = match compare_nat x n with
                                          | Nat_less _ _ _ => patt_bound_evar x
                                          | Nat_equal _ _ _ => ψ
                                          | Nat_greater _ _ _ => patt_bound_evar (pred x)
                                          end.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n, x: db_index
bevar_subst (patt_bound_evar x) ψ n =
match compare_nat x n with
| Nat_less _ _ _ => patt_bound_evar x
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ =>
    patt_bound_evar (Init.Nat.pred x)
end
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n, x: db_index
bevar_subst (patt_bound_evar x) ψ n =
match compare_nat x n with
| Nat_less _ _ _ => patt_bound_evar x
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ =>
    patt_bound_evar (Init.Nat.pred x)
end
    cbn.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n, x: db_index
match compare_nat x n with
| Nat_less _ _ _ => patt_bound_evar x
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred x)
end =
match compare_nat x n with
| Nat_less _ _ _ => patt_bound_evar x
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ =>
    patt_bound_evar (Init.Nat.pred x)
end case_match; done.
  Qed.

  Lemma bevar_subst_bound_svar ψ (pf : well_formed_closed ψ) n X :
    bevar_subst (patt_bound_svar X) ψ n = patt_bound_svar X.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n, X: db_index
bevar_subst (patt_bound_svar X) ψ n =
patt_bound_svar X
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n, X: db_index
bevar_subst (patt_bound_svar X) ψ n =
patt_bound_svar X reflexivity. Qed.

  Lemma bevar_subst_sym ψ (pf : well_formed_closed ψ) n s :
    bevar_subst (patt_sym s) ψ n = patt_sym s.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
s: symbols
bevar_subst (patt_sym s) ψ n = patt_sym s
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
s: symbols
bevar_subst (patt_sym s) ψ n = patt_sym s reflexivity. Qed.

  Lemma bevar_subst_app ψ (pf : well_formed_closed ψ) n ls rs :
    bevar_subst (patt_app ls rs) ψ n = patt_app (bevar_subst ls ψ n) (bevar_subst rs ψ n).
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ls, rs: Pattern
bevar_subst (patt_app ls rs) ψ n =
patt_app (bevar_subst ls ψ n) (bevar_subst rs ψ n)
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ls, rs: Pattern
bevar_subst (patt_app ls rs) ψ n =
patt_app (bevar_subst ls ψ n) (bevar_subst rs ψ n) reflexivity. Qed.

  Lemma bevar_subst_bott ψ (pf : well_formed_closed ψ) n:
    bevar_subst patt_bott ψ n = patt_bott.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
bevar_subst patt_bott ψ n = patt_bott
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
bevar_subst patt_bott ψ n = patt_bott reflexivity. Qed.

  Lemma bevar_subst_imp ψ (pf : well_formed_closed ψ) n ls rs :
    bevar_subst (patt_imp ls rs) ψ n = patt_imp (bevar_subst ls ψ n) (bevar_subst rs ψ n).
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ls, rs: Pattern
bevar_subst (patt_imp ls rs) ψ n =
patt_imp (bevar_subst ls ψ n) (bevar_subst rs ψ n)
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ls, rs: Pattern
bevar_subst (patt_imp ls rs) ψ n =
patt_imp (bevar_subst ls ψ n) (bevar_subst rs ψ n) reflexivity. Qed.

  Lemma bevar_subst_exists ψ (pf : well_formed_closed ψ) n ϕ :
    bevar_subst (patt_exists ϕ) ψ n = patt_exists (bevar_subst ϕ ψ (S n)).
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ϕ: Pattern
bevar_subst (patt_exists ϕ) ψ n =
patt_exists (bevar_subst ϕ ψ (S n))
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ϕ: Pattern
bevar_subst (patt_exists ϕ) ψ n =
patt_exists (bevar_subst ϕ ψ (S n)) reflexivity. Qed.

  Lemma bevar_subst_mu ψ (pf : well_formed_closed ψ) n ϕ :
    bevar_subst (patt_mu ϕ) ψ n = patt_mu (bevar_subst ϕ ψ n).
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ϕ: Pattern
bevar_subst (patt_mu ϕ) ψ n =
patt_mu (bevar_subst ϕ ψ n)
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ϕ: Pattern
bevar_subst (patt_mu ϕ) ψ n =
patt_mu (bevar_subst ϕ ψ n) reflexivity. Qed.

  (* More trivial but useful lemmas *)
  Lemma bsvar_subst_free_evar ψ (pf : well_formed_closed ψ) n x :
    bsvar_subst (patt_free_evar x) ψ n = patt_free_evar x.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
x: evar
bsvar_subst (patt_free_evar x) ψ n = patt_free_evar x
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
x: evar
bsvar_subst (patt_free_evar x) ψ n = patt_free_evar x reflexivity. Qed.

  Lemma bsvar_subst_free_svar ψ (pf : well_formed_closed ψ) n X :
    bsvar_subst (patt_free_svar X) ψ n = patt_free_svar X.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
X: svar
bsvar_subst (patt_free_svar X) ψ n = patt_free_svar X
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
X: svar
bsvar_subst (patt_free_svar X) ψ n = patt_free_svar X reflexivity. Qed.

  Lemma bsvar_subst_bound_evar ψ (pf : well_formed_closed ψ) n x :
    bsvar_subst (patt_bound_evar x) ψ n = patt_bound_evar x.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n, x: db_index
bsvar_subst (patt_bound_evar x) ψ n =
patt_bound_evar x
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n, x: db_index
bsvar_subst (patt_bound_evar x) ψ n =
patt_bound_evar x reflexivity. Qed.

  Lemma bsvar_subst_bound_svar ψ (pf : well_formed_closed ψ) n X :
    bsvar_subst (patt_bound_svar X) ψ n = match compare_nat X n with
                                          | Nat_less _ _ _ => patt_bound_svar X
                                          | Nat_equal _ _ _ => ψ
                                          | Nat_greater _ _ _ => patt_bound_svar (pred X)
                                          end.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n, X: db_index
bsvar_subst (patt_bound_svar X) ψ n =
match compare_nat X n with
| Nat_less _ _ _ => patt_bound_svar X
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ =>
    patt_bound_svar (Init.Nat.pred X)
end
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n, X: db_index
bsvar_subst (patt_bound_svar X) ψ n =
match compare_nat X n with
| Nat_less _ _ _ => patt_bound_svar X
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ =>
    patt_bound_svar (Init.Nat.pred X)
end
    reflexivity.
  Qed.

  Lemma bsvar_subst_sym ψ (pf : well_formed_closed ψ) n s :
    bsvar_subst (patt_sym s) ψ n = patt_sym s.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
s: symbols
bsvar_subst (patt_sym s) ψ n = patt_sym s
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
s: symbols
bsvar_subst (patt_sym s) ψ n = patt_sym s reflexivity. Qed.

  Lemma bsvar_subst_app ψ (pf : well_formed_closed ψ) n ls rs :
    bsvar_subst (patt_app ls rs) ψ n = patt_app (bsvar_subst ls ψ n) (bsvar_subst rs ψ n).
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ls, rs: Pattern
bsvar_subst (patt_app ls rs) ψ n =
patt_app (bsvar_subst ls ψ n) (bsvar_subst rs ψ n)
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ls, rs: Pattern
bsvar_subst (patt_app ls rs) ψ n =
patt_app (bsvar_subst ls ψ n) (bsvar_subst rs ψ n) reflexivity. Qed.

  Lemma bsvar_subst_bott ψ (pf : well_formed_closed ψ) n :
    bsvar_subst patt_bott ψ n = patt_bott.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
bsvar_subst patt_bott ψ n = patt_bott
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
bsvar_subst patt_bott ψ n = patt_bott reflexivity. Qed.

  Lemma bsvar_subst_imp ψ (pf : well_formed_closed ψ) n ls rs:
    bsvar_subst (patt_imp ls rs) ψ n = patt_imp (bsvar_subst ls ψ n) (bsvar_subst rs ψ n).
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ls, rs: Pattern
bsvar_subst (patt_imp ls rs) ψ n =
patt_imp (bsvar_subst ls ψ n) (bsvar_subst rs ψ n)
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ls, rs: Pattern
bsvar_subst (patt_imp ls rs) ψ n =
patt_imp (bsvar_subst ls ψ n) (bsvar_subst rs ψ n) reflexivity. Qed.

  Lemma bsvar_subst_exists ψ (pf : well_formed_closed ψ) n ϕ :
    bsvar_subst (patt_exists ϕ) ψ n = patt_exists (bsvar_subst ϕ ψ n).
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ϕ: Pattern
bsvar_subst (patt_exists ϕ) ψ n =
patt_exists (bsvar_subst ϕ ψ n)
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ϕ: Pattern
bsvar_subst (patt_exists ϕ) ψ n =
patt_exists (bsvar_subst ϕ ψ n) reflexivity. Qed.

  Lemma bsvar_subst_mu ψ (pf : well_formed_closed ψ) n ϕ :
    bsvar_subst (patt_mu ϕ) ψ n = patt_mu (bsvar_subst ϕ ψ (S n)).
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ϕ: Pattern
bsvar_subst (patt_mu ϕ) ψ n =
patt_mu (bsvar_subst ϕ ψ (S n))
  Proof.
Σ: Signature
ψ: Pattern
pf: well_formed_closed ψ
n: db_index
ϕ: Pattern
bsvar_subst (patt_mu ϕ) ψ n =
patt_mu (bsvar_subst ϕ ψ (S n)) reflexivity. Qed.


End subst.

Module Notations.

Declare Scope ml_scope.
Notation "e ^[ 'evar:' dbi ↦ e' ]" := (bevar_subst e e' dbi) (at level 2, e' at level 200, left associativity,
format "e ^[ 'evar:' dbi ↦ e' ]" ) : ml_scope.
Notation "e ^[ 'svar:' dbi ↦ e' ]" := (bsvar_subst e e' dbi) (at level 2, e' at level 200, left associativity,
format "e ^[ 'svar:' dbi ↦ e' ]" ) : ml_scope.
Notation "e ^[[ 'evar:' x ↦ e' ]]" := (free_evar_subst e e' x) (at level 2, e' at level 200, left associativity,
format "e ^[[ 'evar:' x ↦ e' ]]" ) : ml_scope.
Notation "e ^[[ 'svar:' X ↦ e' ]]" := (free_svar_subst e e' X) (at level 2, e' at level 200, left associativity,
format "e ^[[ 'svar:' X ↦ e' ]]" ) : ml_scope.
Notation "e ^ [ e' ]" := (instantiate e e') (at level 2, e' at level 200, left associativity) : ml_scope.

End Notations.

Import Notations.
Open Scope ml_scope.

Section subst.
    Context {Σ : Signature}.


Lemma wfc_ex_aux_bevar_subst :
forall phi psi n,
  well_formed_closed_ex_aux phi (S n) = true
  -> well_formed_closed_ex_aux psi n = true
  -> well_formed_closed_ex_aux (bevar_subst phi psi n) n = true.
Σ: Signature
∀ (phi psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[evar:n↦psi] n =
      true
Proof.
Σ: Signature
∀ (phi psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[evar:n↦psi] n =
      true
intros phi psi n H H0.
Σ: Signature
phi, psi: Pattern
n: nat
H: well_formed_closed_ex_aux phi (S n) = true
H0: well_formed_closed_ex_aux psi n = true
well_formed_closed_ex_aux phi^[evar:n↦psi] n = true 
generalize dependent n.
Σ: Signature
phi, psi: Pattern
∀ n : nat,
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[evar:n↦psi] n =
      true generalize dependent psi.
Σ: Signature
phi: Pattern
∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[evar:n↦psi] n =
      true
induction phi; intros psi n' H H0; try lia; auto.
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_bound_evar n)
  (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_bound_evar n)^[evar:n'↦psi] n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_app phi1 phi2)^[evar:n'↦psi] n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^[evar:n'↦psi] n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_exists phi) (S n') =
true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_exists phi)^[evar:n'↦psi] n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_mu phi) (S n') =
true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux 
  (patt_mu phi)^[evar:n'↦psi] n' = true
-
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_bound_evar n)
  (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_bound_evar n)^[evar:n'↦psi] n' = true simpl in *.
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n' = true unfold well_formed_closed_ex_aux.
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_ex_aux psi n' = true
(fix well_formed_closed_ex_aux
   (phi : Pattern) (max_ind_evar : db_index) {struct
    phi} : bool :=
   match phi with
   | patt_bound_evar n =>
       if decide (n < max_ind_evar)
       then true
       else false
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_closed_ex_aux psi1 max_ind_evar &&
       well_formed_closed_ex_aux psi2 max_ind_evar
   | patt_exists psi =>
       well_formed_closed_ex_aux psi (S max_ind_evar)
   | patt_mu psi =>
       well_formed_closed_ex_aux psi max_ind_evar
   | _ => true
   end)
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n' = true repeat case_match; simpl; auto.
Σ: Signature
n: db_index
psi: Pattern
n': nat
l: n < S n'
H1: decide (n < S n') = left l
H: true = true
H0: well_formed_closed_ex_aux psi n' = true
g: n > n'
H2: compare_nat n n' = Nat_greater n n' g
n0: ¬ Nat.pred n < n'
H3: decide (Nat.pred n < n') = right n0
false = true lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_app phi1 phi2)^[evar:n'↦psi] n' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi1 (S n') &&
well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi1 (S n') = true
∧ well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true
  rewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true rewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi2 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true && true = true apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true && true = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true && true = true
  reflexivity.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^[evar:n'↦psi] n' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi1 (S n') &&
well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi1 (S n') = true
∧ well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n' &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true
  rewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi1 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n' = true rewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi2 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true && true = true apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true && true = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi1 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[evar:n↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi2 (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[evar:n↦psi] n = true
psi: Pattern
n': nat
H1: well_formed_closed_ex_aux phi1 (S n') = true
H2: well_formed_closed_ex_aux phi2 (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true && true = true
  reflexivity.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_exists phi) (S n') =
true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux
  (patt_exists phi)^[evar:n'↦psi] n' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_exists phi) (S n') =
true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi^[evar:S n'↦psi] (S n') =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi^[evar:S n'↦psi] (S n') =
true rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi (S (S n')) = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
true = true assumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
true = true 2: reflexivity.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi (S n') = true
  eapply well_formed_closed_ex_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
?ind_evar1 ≤ S n'
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi ?ind_evar1 = true 2: apply H0.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S (S n')) = true
H0: well_formed_closed_ex_aux psi n' = true
n' ≤ S n' lia.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_mu phi) (S n') =
true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux (patt_mu phi)^[evar:n'↦psi]
  n' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_mu phi) (S n') =
true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi^[evar:n'↦psi] n' = true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi^[evar:n'↦psi] n' = true
  rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux phi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true = true apply H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true = true
  eapply well_formed_closed_ex_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
?ind_evar1 ≤ n'
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
well_formed_closed_ex_aux psi ?ind_evar1 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true = true 2: apply H0.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
n' ≤ n'
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true = true lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n : nat),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[evar:n↦psi] n = true
psi: Pattern
n': nat
H: well_formed_closed_ex_aux phi (S n') = true
H0: well_formed_closed_ex_aux psi n' = true
true = true
  reflexivity.
Qed.

Lemma wfc_mu_aux_bevar_subst :
forall phi psi n n',
  well_formed_closed_mu_aux phi n' = true
  -> well_formed_closed_mu_aux psi n' = true
  -> well_formed_closed_mu_aux (bevar_subst phi psi n) n' = true.
Σ: Signature
∀ (phi psi : Pattern) (n n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux phi^[evar:n↦psi] n' =
      true
Proof.
Σ: Signature
∀ (phi psi : Pattern) (n n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux phi^[evar:n↦psi] n' =
      true
intros phi psi n n' H H0.
Σ: Signature
phi, psi: Pattern
n, n': db_index
H: well_formed_closed_mu_aux phi n' = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi^[evar:n↦psi] n' = true 
generalize dependent n.
Σ: Signature
phi, psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
H0: well_formed_closed_mu_aux psi n' = true
∀ n : db_index,
  well_formed_closed_mu_aux phi^[evar:n↦psi] n' = true generalize dependent n'.
Σ: Signature
phi, psi: Pattern
∀ n' : db_index,
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux phi^[evar:n↦psi] n' =
        true generalize dependent psi.
Σ: Signature
phi: Pattern
∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux phi^[evar:n↦psi] n' =
        true
induction phi; intros psi n' H n'' H0; try lia; auto.
Σ: Signature
n: db_index
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_bound_evar n) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_bound_evar n)^[evar:H0↦psi] n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_app phi1 phi2) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_app phi1 phi2)^[evar:H0↦psi] n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_imp phi1 phi2) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^[evar:H0↦psi] n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_exists phi) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_exists phi)^[evar:H0↦psi] n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_mu phi) n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux 
  (patt_mu phi)^[evar:H0↦psi] n' = true
-
Σ: Signature
n: db_index
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_bound_evar n) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_bound_evar n)^[evar:H0↦psi] n' = true simpl in *.
Σ: Signature
n: db_index
psi: Pattern
n': db_index
H: true = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  match compare_nat n H0 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n' = true unfold well_formed_closed_mu_aux.
Σ: Signature
n: db_index
psi: Pattern
n': db_index
H: true = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
(fix well_formed_closed_mu_aux
   (phi : Pattern) (max_ind_svar : db_index) {struct
    phi} : bool :=
   match phi with
   | patt_bound_svar n =>
       if decide (n < max_ind_svar)
       then true
       else false
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_closed_mu_aux psi1 max_ind_svar &&
       well_formed_closed_mu_aux psi2 max_ind_svar
   | patt_exists psi =>
       well_formed_closed_mu_aux psi max_ind_svar
   | patt_mu psi =>
       well_formed_closed_mu_aux psi (S max_ind_svar)
   | _ => true
   end)
  match compare_nat n H0 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n' = true repeat case_match; simpl; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_app phi1 phi2) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_app phi1 phi2)^[evar:H0↦psi] n' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_app phi1 phi2) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1^[evar:H0↦psi] n' &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1^[evar:H0↦psi] n' &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true
  rewrite IHphi1; auto.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1 n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
true &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true 2: rewrite IHphi2; auto.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1 n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi2 n' = true all: destruct_and!; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_imp phi1 phi2) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^[evar:H0↦psi] n' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_imp phi1 phi2) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1^[evar:H0↦psi] n' &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1^[evar:H0↦psi] n' &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H1: well_formed_closed_mu_aux phi1 n' = true
H2: well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi1^[evar:H0↦psi] n' &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true
  rewrite IHphi1; auto.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi1 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi1^[evar:n↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi2 n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi2^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H1: well_formed_closed_mu_aux phi1 n' = true
H2: well_formed_closed_mu_aux phi2 n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
true &&
well_formed_closed_mu_aux phi2^[evar:H0↦psi] n' = true rewrite IHphi2; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_exists phi) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux
  (patt_exists phi)^[evar:H0↦psi] n' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_exists phi) n' =
true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi^[evar:S H0↦psi] n' =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi^[evar:S H0↦psi] n' =
true rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
true = true assumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
true = true 2: reflexivity.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi n' = true
  eauto using well_formed_closed_ex_aux_ind.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_mu phi) n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux (patt_mu phi)^[evar:H0↦psi]
  n' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux (patt_mu phi) n' = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi^[evar:H0↦psi] (S n') =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi^[evar:H0↦psi] (S n') =
true
  rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux phi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
true = true apply H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
true = true 2: reflexivity.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi (S n') = true
  eapply well_formed_closed_mu_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
?ind_svar1 ≤ S n'
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
well_formed_closed_mu_aux psi ?ind_svar1 = true
  2: eassumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : db_index),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux psi n' = true
    → ∀ n : db_index,
        well_formed_closed_mu_aux
          phi^[evar:n↦psi] n' = true
psi: Pattern
n': db_index
H: well_formed_closed_mu_aux phi (S n') = true
n'': well_formed_closed_mu_aux psi n' = true
H0: db_index
n' ≤ S n' lia.
Qed.


Lemma wfc_ex_aux_bsvar_subst :
forall phi psi n n',
  well_formed_closed_ex_aux phi n = true
  -> well_formed_closed_ex_aux psi n = true
  -> well_formed_closed_ex_aux (bsvar_subst phi psi n') n = true.
Σ: Signature
∀ (phi psi : Pattern) (n n' : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[svar:n'↦psi] n =
      true
Proof.
Σ: Signature
∀ (phi psi : Pattern) (n n' : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[svar:n'↦psi] n =
      true
intros phi psi n n' H H0.
Σ: Signature
phi, psi: Pattern
n, n': db_index
H: well_formed_closed_ex_aux phi n = true
H0: well_formed_closed_ex_aux psi n = true
well_formed_closed_ex_aux phi^[svar:n'↦psi] n = true 
generalize dependent n.
Σ: Signature
phi, psi: Pattern
n': db_index
∀ n : db_index,
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[svar:n'↦psi] n =
      true generalize dependent n'.
Σ: Signature
phi, psi: Pattern
∀ n' n : db_index,
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[svar:n'↦psi] n =
      true generalize dependent psi.
Σ: Signature
phi: Pattern
∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux phi^[svar:n'↦psi] n =
      true
induction phi; intros psi n' n'' H H0; try lia; auto.
Σ: Signature
n: db_index
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_bound_svar n) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_bound_svar n)^[svar:n'↦psi] n'' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_app phi1 phi2) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_app phi1 phi2)^[svar:n'↦psi] n'' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_imp phi1 phi2) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^[svar:n'↦psi] n'' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_exists phi) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_exists phi)^[svar:n'↦psi] n'' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_mu phi) n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux 
  (patt_mu phi)^[svar:n'↦psi] n'' = true
-
Σ: Signature
n: db_index
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_bound_svar n) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_bound_svar n)^[svar:n'↦psi] n'' = true simpl in *.
Σ: Signature
n: db_index
psi: Pattern
n', n'': db_index
H: true = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end n'' = true unfold well_formed_closed_ex_aux.
Σ: Signature
n: db_index
psi: Pattern
n', n'': db_index
H: true = true
H0: well_formed_closed_ex_aux psi n'' = true
(fix well_formed_closed_ex_aux
   (phi : Pattern) (max_ind_evar : db_index) {struct
    phi} : bool :=
   match phi with
   | patt_bound_evar n =>
       if decide (n < max_ind_evar)
       then true
       else false
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_closed_ex_aux psi1 max_ind_evar &&
       well_formed_closed_ex_aux psi2 max_ind_evar
   | patt_exists psi =>
       well_formed_closed_ex_aux psi (S max_ind_evar)
   | patt_mu psi =>
       well_formed_closed_ex_aux psi max_ind_evar
   | _ => true
   end)
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end n'' = true repeat case_match; simpl; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_app phi1 phi2) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_app phi1 phi2)^[svar:n'↦psi] n'' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_app phi1 phi2) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi1^[svar:n'↦psi] n'' &&
well_formed_closed_ex_aux phi2^[svar:n'↦psi] n'' =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi1 n'' &&
well_formed_closed_ex_aux phi2 n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi1^[svar:n'↦psi] n'' &&
well_formed_closed_ex_aux phi2^[svar:n'↦psi] n'' =
true destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H1: well_formed_closed_ex_aux phi1 n'' = true
H2: well_formed_closed_ex_aux phi2 n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi1^[svar:n'↦psi] n'' &&
well_formed_closed_ex_aux phi2^[svar:n'↦psi] n'' =
true split_and; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_imp phi1 phi2) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^[svar:n'↦psi] n'' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_imp phi1 phi2) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi1^[svar:n'↦psi] n'' &&
well_formed_closed_ex_aux phi2^[svar:n'↦psi] n'' =
true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi1 n'' &&
well_formed_closed_ex_aux phi2 n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi1^[svar:n'↦psi] n'' &&
well_formed_closed_ex_aux phi2^[svar:n'↦psi] n'' =
true destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi1 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi2 n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi2^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H1: well_formed_closed_ex_aux phi1 n'' = true
H2: well_formed_closed_ex_aux phi2 n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi1^[svar:n'↦psi] n'' &&
well_formed_closed_ex_aux phi2^[svar:n'↦psi] n'' =
true split_and; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_exists phi) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux
  (patt_exists phi)^[svar:n'↦psi] n'' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_exists phi) n'' =
true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi^[svar:n'↦psi] (S n'') =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi^[svar:n'↦psi] (S n'') =
true rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi (S n'') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux psi (S n'') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true assumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux psi (S n'') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true
  eapply well_formed_closed_ex_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
?ind_evar1 ≤ S n''
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux psi ?ind_evar1 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true 2: eassumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
n'' ≤ S n''
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi (S n'') = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true reflexivity.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_mu phi) n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux (patt_mu phi)^[svar:n'↦psi]
  n'' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux (patt_mu phi) n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi^[svar:S n'↦psi] n'' =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi^[svar:S n'↦psi] n'' =
true
  rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux phi n'' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux psi n'' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true apply H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux psi n'' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true
  eapply well_formed_closed_ex_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
?ind_evar1 ≤ n''
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
well_formed_closed_ex_aux psi ?ind_evar1 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true 2: eassumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
n'' ≤ n''
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' n : db_index),
  well_formed_closed_ex_aux phi n = true
  → well_formed_closed_ex_aux psi n = true
    → well_formed_closed_ex_aux
        phi^[svar:n'↦psi] n = true
psi: Pattern
n', n'': db_index
H: well_formed_closed_ex_aux phi n'' = true
H0: well_formed_closed_ex_aux psi n'' = true
true = true reflexivity.
Qed.

Lemma wfc_mu_aux_bsvar_subst :
forall phi psi n',
  well_formed_closed_mu_aux phi (S n') = true
  -> well_formed_closed_mu_aux psi n' = true
  -> well_formed_closed_mu_aux (bsvar_subst phi psi n') n' = true.
Σ: Signature
∀ (phi psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux phi^[svar:n'↦psi] n' =
      true
Proof.
Σ: Signature
∀ (phi psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux phi^[svar:n'↦psi] n' =
      true
intros phi psi n' H H0.
Σ: Signature
phi, psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi^[svar:n'↦psi] n' = true 
generalize dependent n'.
Σ: Signature
phi, psi: Pattern
∀ n' : nat,
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux phi^[svar:n'↦psi] n' =
      true generalize dependent psi.
Σ: Signature
phi: Pattern
∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux phi^[svar:n'↦psi] n' =
      true
induction phi; intros psi n' H H0; try lia; auto.
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_bound_svar n)
  (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_bound_svar n)^[svar:n'↦psi] n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_app phi1 phi2)^[svar:n'↦psi] n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^[svar:n'↦psi] n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_exists phi) (S n') =
true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_exists phi)^[svar:n'↦psi] n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_mu phi) (S n') =
true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux 
  (patt_mu phi)^[svar:n'↦psi] n' = true
-
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_bound_svar n)
  (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_bound_svar n)^[svar:n'↦psi] n' = true simpl in *.
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end n' = true unfold well_formed_closed_mu_aux.
Σ: Signature
n: db_index
psi: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_mu_aux psi n' = true
(fix well_formed_closed_mu_aux
   (phi : Pattern) (max_ind_svar : db_index) {struct
    phi} : bool :=
   match phi with
   | patt_bound_svar n =>
       if decide (n < max_ind_svar)
       then true
       else false
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_closed_mu_aux psi1 max_ind_svar &&
       well_formed_closed_mu_aux psi2 max_ind_svar
   | patt_exists psi =>
       well_formed_closed_mu_aux psi max_ind_svar
   | patt_mu psi =>
       well_formed_closed_mu_aux psi (S max_ind_svar)
   | _ => true
   end)
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end n' = true repeat case_match; simpl; auto.
Σ: Signature
n: db_index
psi: Pattern
n': nat
l: n < S n'
H1: decide (n < S n') = left l
H: true = true
H0: well_formed_closed_mu_aux psi n' = true
g: n > n'
H2: compare_nat n n' = Nat_greater n n' g
n0: ¬ Nat.pred n < n'
H3: decide (Nat.pred n < n') = right n0
false = true lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_app phi1 phi2)^[svar:n'↦psi] n' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n' &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi1 (S n') &&
well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n' &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n' &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true
  rewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true rewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi2 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true && true = true apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true && true = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true && true = true
  reflexivity.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^[svar:n'↦psi] n' = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n' &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi1 (S n') &&
well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n' &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n' &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true
  rewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi1 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n' = true rewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi2 (S n') = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true && true = true apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true && true = true assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi1 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi1^[svar:n'↦psi] n' = true
IHphi2: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi2 (S n') =
  true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi2^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H1: well_formed_closed_mu_aux phi1 (S n') = true
H2: well_formed_closed_mu_aux phi2 (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true && true = true
  reflexivity.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_exists phi) (S n') =
true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux
  (patt_exists phi)^[svar:n'↦psi] n' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_exists phi) (S n') =
true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi^[svar:n'↦psi] n' = true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi^[svar:n'↦psi] n' = true rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true = true assumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi n' = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true = true
  assumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S n') = true
H0: well_formed_closed_mu_aux psi n' = true
true = true reflexivity.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_mu phi) (S n') =
true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux (patt_mu phi)^[svar:n'↦psi]
  n' = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux (patt_mu phi) (S n') =
true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi^[svar:S n'↦psi] (S n') =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi^[svar:S n'↦psi] (S n') =
true
  rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux phi (S (S n')) = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
true = true apply H.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi (S n') = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
true = true
  eapply well_formed_closed_mu_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
?ind_svar1 ≤ S n'
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
well_formed_closed_mu_aux psi ?ind_svar1 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
true = true 2: eassumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
n' ≤ S n'
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
true = true lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n' : nat),
  well_formed_closed_mu_aux phi (S n') = true
  → well_formed_closed_mu_aux psi n' = true
    → well_formed_closed_mu_aux
        phi^[svar:n'↦psi] n' = true
psi: Pattern
n': nat
H: well_formed_closed_mu_aux phi (S (S n')) = true
H0: well_formed_closed_mu_aux psi n' = true
true = true reflexivity.
Qed.


(*Helper lemma for wf_ex_to_wf_body *)
Corollary wfc_ex_aux_body_ex_imp1:
forall phi n x,
  well_formed_closed_ex_aux phi (S n) = true
  ->
  well_formed_closed_ex_aux (evar_open n x phi) n = true.
Σ: Signature
∀ (phi : Pattern) (n : nat) (x : evar),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux (evar_open n x phi) n =
    true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n : nat) (x : evar),
  well_formed_closed_ex_aux phi (S n) = true
  → well_formed_closed_ex_aux (evar_open n x phi) n =
    true
intros.
Σ: Signature
phi: Pattern
n: nat
x: evar
H: well_formed_closed_ex_aux phi (S n) = true
well_formed_closed_ex_aux (evar_open n x phi) n = true apply wfc_ex_aux_bevar_subst; auto.
Qed.

Corollary wfc_mu_aux_body_ex_imp1:
forall phi n n' x,
  well_formed_closed_mu_aux phi n' = true
  ->
  well_formed_closed_mu_aux (evar_open n x phi) n' = true.
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux (evar_open n x phi) n' =
    true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (x : evar),
  well_formed_closed_mu_aux phi n' = true
  → well_formed_closed_mu_aux (evar_open n x phi) n' =
    true
intros.
Σ: Signature
phi: Pattern
n, n': db_index
x: evar
H: well_formed_closed_mu_aux phi n' = true
well_formed_closed_mu_aux (evar_open n x phi) n' =
true now apply wfc_mu_aux_bevar_subst.
Qed.

Corollary wfc_ex_aux_body_mu_imp1:
forall phi n n' X,
  well_formed_closed_ex_aux phi n' = true
  ->
  well_formed_closed_ex_aux (svar_open n X phi) n' = true.
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi n' = true
  → well_formed_closed_ex_aux (svar_open n X phi) n' =
    true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n n' : db_index) (X : svar),
  well_formed_closed_ex_aux phi n' = true
  → well_formed_closed_ex_aux (svar_open n X phi) n' =
    true
intros.
Σ: Signature
phi: Pattern
n, n': db_index
X: svar
H: well_formed_closed_ex_aux phi n' = true
well_formed_closed_ex_aux (svar_open n X phi) n' =
true now apply wfc_ex_aux_bsvar_subst.
Qed.

Corollary wfc_mu_aux_body_mu_imp1:
forall phi n X,
  well_formed_closed_mu_aux phi (S n) = true
  ->
  well_formed_closed_mu_aux (svar_open n X phi) n = true.
Σ: Signature
∀ (phi : Pattern) (n : nat) (X : svar),
  well_formed_closed_mu_aux phi (S n) = true
  → well_formed_closed_mu_aux (svar_open n X phi) n =
    true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n : nat) (X : svar),
  well_formed_closed_mu_aux phi (S n) = true
  → well_formed_closed_mu_aux (svar_open n X phi) n =
    true
intros.
Σ: Signature
phi: Pattern
n: nat
X: svar
H: well_formed_closed_mu_aux phi (S n) = true
well_formed_closed_mu_aux (svar_open n X phi) n = true now apply wfc_mu_aux_bsvar_subst.
Qed.

Lemma wfc_mu_aux_bsvar_subst_le :
forall phi psi n n', n' <= n ->
  well_formed_closed_mu_aux phi (S n) = true ->
  well_formed_closed_mu_aux psi n
  ->
  well_formed_closed_mu_aux (phi^[svar:n' ↦ psi]) n = true.
Σ: Signature
∀ (phi psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux phi^[svar:n'↦psi] n =
        true
Proof using .
Σ: Signature
∀ (phi psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux phi^[svar:n'↦psi] n =
        true
induction phi; intros psi n0 n' H Hwf1 Hwf2; try lia; auto.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_bound_svar n)^[svar:n'↦psi] n0 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_app phi1 phi2)^[svar:n'↦psi] n0 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^[svar:n'↦psi] n0 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_exists phi)^[svar:n'↦psi] n0 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux (patt_mu phi) (S n0) =
true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux 
  (patt_mu phi)^[svar:n'↦psi] n0 = true
-
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_bound_svar n)^[svar:n'↦psi] n0 = true simpl.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end n0 = true case_match; auto.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
l: n < n'
H0: compare_nat n n' = Nat_less n n' l
well_formed_closed_mu_aux (patt_bound_svar n) n0 =
true
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_mu_aux
  (patt_bound_svar (Nat.pred n)) n0 = true simpl.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
l: n < n'
H0: compare_nat n n' = Nat_less n n' l
(if decide (n < n0) then true else false) = true
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_mu_aux
  (patt_bound_svar (Nat.pred n)) n0 = true case_match; try lia.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_bound_svar n) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_mu_aux
  (patt_bound_svar (Nat.pred n)) n0 = true
  simpl in Hwf1.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: (if decide (n < S n0) then true else false) =
true
Hwf2: well_formed_closed_mu_aux psi n0
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_mu_aux
  (patt_bound_svar (Nat.pred n)) n0 = true case_match; try lia.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
l: n < S n0
H1: decide (n < S n0) = left l
Hwf1: true = true
Hwf2: well_formed_closed_mu_aux psi n0
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_mu_aux
  (patt_bound_svar (Nat.pred n)) n0 = true simpl.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
l: n < S n0
H1: decide (n < S n0) = left l
Hwf1: true = true
Hwf2: well_formed_closed_mu_aux psi n0
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
(if decide (Nat.pred n < n0) then true else false) =
true case_match; lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_app phi1 phi2)^[svar:n'↦psi] n0 = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true simpl in Hwf1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi1 (S n0) &&
well_formed_closed_mu_aux phi2 (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true apply andb_true_iff in Hwf1 as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_mu_aux phi1 (S n0) = true
H2: well_formed_closed_mu_aux phi2 (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true
  rewrite (IHphi1 _ _ n'); auto.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_mu_aux phi1 (S n0) = true
H2: well_formed_closed_mu_aux phi2 (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true rewrite (IHphi2 _ _ n'); auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_imp phi1 phi2)^[svar:n'↦psi] n0 = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true simpl in Hwf1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi1 (S n0) &&
well_formed_closed_mu_aux phi2 (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true apply andb_true_iff in Hwf1 as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_mu_aux phi1 (S n0) = true
H2: well_formed_closed_mu_aux phi2 (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi1^[svar:n'↦psi] n0 &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true
  rewrite (IHphi1 _ _ n'); auto.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi1 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi1^[svar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi2 (S n) =
    true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi2^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_mu_aux phi1 (S n0) = true
H2: well_formed_closed_mu_aux phi2 (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
true &&
well_formed_closed_mu_aux phi2^[svar:n'↦psi] n0 = true rewrite (IHphi2 _ _ n'); auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux
  (patt_exists phi)^[svar:n'↦psi] n0 = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi^[svar:n'↦psi] n0 = true simpl in Hwf1.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S n0) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi^[svar:n'↦psi] n0 = true rewrite (IHphi _ _ n'); auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux (patt_mu phi) (S n0) =
true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux (patt_mu phi)^[svar:n'↦psi]
  n0 = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux (patt_mu phi) (S n0) =
true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi^[svar:S n'↦psi] (S n0) =
true simpl in Hwf1.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux phi^[svar:S n'↦psi] (S n0) =
true rewrite (IHphi _ _ (S n')); auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
S n' ≤ S n0
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux psi (S n0) lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux psi (S n0)
  eapply well_formed_closed_mu_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
?ind_svar1 ≤ S n0
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
well_formed_closed_mu_aux psi ?ind_svar1 = true 2: eassumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux psi n
      → well_formed_closed_mu_aux
          phi^[svar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_mu_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_mu_aux psi n0
n0 ≤ S n0 lia.
Qed.

Lemma wfc_ex_aux_bsvar_subst_le:
forall phi psi n n', n' <= n ->
  well_formed_closed_ex_aux phi (S n) = true ->
  well_formed_closed_ex_aux psi n = true
  ->
  well_formed_closed_ex_aux (phi^[evar:n'↦psi]) n = true.
Σ: Signature
∀ (phi psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux phi^[evar:n'↦psi] n =
        true
Proof using .
Σ: Signature
∀ (phi psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux phi^[evar:n'↦psi] n =
        true
induction phi; intros psi n0 n' H Hwf1 Hwf2; try lia; auto.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_bound_evar n)^[evar:n'↦psi] n0 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_app phi1 phi2)^[evar:n'↦psi] n0 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^[evar:n'↦psi] n0 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_exists phi)^[evar:n'↦psi] n0 = true
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux (patt_mu phi) (S n0) =
true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux 
  (patt_mu phi)^[evar:n'↦psi] n0 = true
-
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_bound_evar n)^[evar:n'↦psi] n0 = true simpl.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n0 = true case_match; auto.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
l: n < n'
H0: compare_nat n n' = Nat_less n n' l
well_formed_closed_ex_aux (patt_bound_evar n) n0 =
true
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) n0 = true simpl.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
l: n < n'
H0: compare_nat n n' = Nat_less n n' l
(if decide (n < n0) then true else false) = true
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) n0 = true case_match; try lia.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_bound_evar n) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) n0 = true
  simpl in Hwf1.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: (if decide (n < S n0) then true else false) =
true
Hwf2: well_formed_closed_ex_aux psi n0 = true
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) n0 = true case_match; try lia.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
l: n < S n0
H1: decide (n < S n0) = left l
Hwf1: true = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) n0 = true simpl.
Σ: Signature
n: db_index
psi: Pattern
n0, n': nat
H: n' ≤ n0
l: n < S n0
H1: decide (n < S n0) = left l
Hwf1: true = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
g: n > n'
H0: compare_nat n n' = Nat_greater n n' g
(if decide (Nat.pred n < n0) then true else false) =
true case_match; lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_app phi1 phi2)^[evar:n'↦psi] n0 = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true simpl in Hwf1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi1 (S n0) &&
well_formed_closed_ex_aux phi2 (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true apply andb_true_iff in Hwf1 as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_ex_aux phi1 (S n0) = true
H2: well_formed_closed_ex_aux phi2 (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true
  rewrite (IHphi1 _ _ n'); auto.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_ex_aux phi1 (S n0) = true
H2: well_formed_closed_ex_aux phi2 (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true rewrite (IHphi2 _ _ n'); auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_imp phi1 phi2)^[evar:n'↦psi] n0 = true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true simpl in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true simpl in Hwf1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi1 (S n0) &&
well_formed_closed_ex_aux phi2 (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true apply andb_true_iff in Hwf1 as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_ex_aux phi1 (S n0) = true
H2: well_formed_closed_ex_aux phi2 (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi1^[evar:n'↦psi] n0 &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true
  rewrite (IHphi1 _ _ n'); auto.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi1 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi1^[evar:n'↦psi] n = true
IHphi2: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi2 (S n) =
    true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi2^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
H1: well_formed_closed_ex_aux phi1 (S n0) = true
H2: well_formed_closed_ex_aux phi2 (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
true &&
well_formed_closed_ex_aux phi2^[evar:n'↦psi] n0 = true rewrite (IHphi2 _ _ n'); auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux
  (patt_exists phi)^[evar:n'↦psi] n0 = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi^[evar:S n'↦psi] (S n0) =
true simpl in Hwf1.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi^[evar:S n'↦psi] (S n0) =
true rewrite (IHphi _ _ (S n')); auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
S n' ≤ S n0
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux psi (S n0) = true lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux psi (S n0) = true
  eapply well_formed_closed_ex_aux_ind.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
?ind_evar1 ≤ S n0
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux psi ?ind_evar1 = true 2: eassumption.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S (S n0)) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
n0 ≤ S n0 lia.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux (patt_mu phi) (S n0) =
true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux (patt_mu phi)^[evar:n'↦psi]
  n0 = true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux (patt_mu phi) (S n0) =
true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi^[evar:n'↦psi] n0 = true simpl in Hwf1.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi : Pattern) (n n' : nat),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux psi n = true
      → well_formed_closed_ex_aux
          phi^[evar:n'↦psi] n = true
psi: Pattern
n0, n': nat
H: n' ≤ n0
Hwf1: well_formed_closed_ex_aux phi (S n0) = true
Hwf2: well_formed_closed_ex_aux psi n0 = true
well_formed_closed_ex_aux phi^[evar:n'↦psi] n0 = true rewrite (IHphi _ _ n'); auto.
Qed.

Corollary wfc_mu_aux_body_mu_imp3:
forall phi n n' X, n' <= n ->
  well_formed_closed_mu_aux phi (S n) = true
  ->
  well_formed_closed_mu_aux (svar_open n' X phi) n = true.
Σ: Signature
∀ (phi : Pattern) (n n' : nat) (X : svar),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux (svar_open n' X phi) n =
      true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n n' : nat) (X : svar),
  n' ≤ n
  → well_formed_closed_mu_aux phi (S n) = true
    → well_formed_closed_mu_aux (svar_open n' X phi) n =
      true
intros.
Σ: Signature
phi: Pattern
n, n': nat
X: svar
H: n' ≤ n
H0: well_formed_closed_mu_aux phi (S n) = true
well_formed_closed_mu_aux (svar_open n' X phi) n =
true now apply wfc_mu_aux_bsvar_subst_le.
Qed.

Corollary wfc_mu_aux_body_ex_imp3:
forall phi n n' X, n' <= n ->
  well_formed_closed_ex_aux phi (S n) = true
  ->
  well_formed_closed_ex_aux (evar_open n' X phi) n = true.
Σ: Signature
∀ (phi : Pattern) (n n' : nat) (X : evar),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux (evar_open n' X phi) n =
      true
Proof using .
Σ: Signature
∀ (phi : Pattern) (n n' : nat) (X : evar),
  n' ≤ n
  → well_formed_closed_ex_aux phi (S n) = true
    → well_formed_closed_ex_aux (evar_open n' X phi) n =
      true
intros.
Σ: Signature
phi: Pattern
n, n': nat
X: evar
H: n' ≤ n
H0: well_formed_closed_ex_aux phi (S n) = true
well_formed_closed_ex_aux (evar_open n' X phi) n =
true now apply wfc_ex_aux_bsvar_subst_le.
Qed.

Corollary wfc_ex_aux_body_iff: 
forall phi n x,
  well_formed_closed_ex_aux phi (S n) = true
  <->
  well_formed_closed_ex_aux (evar_open n x phi) n = true.
Σ: Signature
∀ (phi : Pattern) (n : nat) (x : evar),
  well_formed_closed_ex_aux phi (S n) = true
  ↔ well_formed_closed_ex_aux (evar_open n x phi) n =
    true
Proof.
Σ: Signature
∀ (phi : Pattern) (n : nat) (x : evar),
  well_formed_closed_ex_aux phi (S n) = true
  ↔ well_formed_closed_ex_aux (evar_open n x phi) n =
    true
split.
Σ: Signature
phi: Pattern
n: nat
x: evar
well_formed_closed_ex_aux phi (S n) = true
→ well_formed_closed_ex_aux (evar_open n x phi) n =
  true
Σ: Signature
phi: Pattern
n: nat
x: evar
well_formed_closed_ex_aux (evar_open n x phi) n = true
→ well_formed_closed_ex_aux phi (S n) = true
apply wfc_ex_aux_body_ex_imp1.
Σ: Signature
phi: Pattern
n: nat
x: evar
well_formed_closed_ex_aux (evar_open n x phi) n = true
→ well_formed_closed_ex_aux phi (S n) = true
apply wfc_ex_aux_body_ex_imp2.
Qed.

Corollary wfc_mu_aux_body_iff: 
forall phi n X,
  well_formed_closed_mu_aux phi (S n) = true
  <->
  well_formed_closed_mu_aux (svar_open n X phi) n = true.
Σ: Signature
∀ (phi : Pattern) (n : nat) (X : svar),
  well_formed_closed_mu_aux phi (S n) = true
  ↔ well_formed_closed_mu_aux (svar_open n X phi) n =
    true
Proof.
Σ: Signature
∀ (phi : Pattern) (n : nat) (X : svar),
  well_formed_closed_mu_aux phi (S n) = true
  ↔ well_formed_closed_mu_aux (svar_open n X phi) n =
    true
split.
Σ: Signature
phi: Pattern
n: nat
X: svar
well_formed_closed_mu_aux phi (S n) = true
→ well_formed_closed_mu_aux (svar_open n X phi) n =
  true
Σ: Signature
phi: Pattern
n: nat
X: svar
well_formed_closed_mu_aux (svar_open n X phi) n = true
→ well_formed_closed_mu_aux phi (S n) = true
apply wfc_mu_aux_body_mu_imp1.
Σ: Signature
phi: Pattern
n: nat
X: svar
well_formed_closed_mu_aux (svar_open n X phi) n = true
→ well_formed_closed_mu_aux phi (S n) = true
apply wfc_mu_aux_body_mu_imp2.
Qed.


(*If (ex, phi) is closed, then its body is closed too*)
Corollary wfc_ex_to_wfc_body:
forall phi, well_formed_closed (patt_exists phi) = true -> wfc_body_ex phi.
Σ: Signature
∀ phi : Pattern,
  well_formed_closed (patt_exists phi) = true
  → wfc_body_ex phi
Proof.
Σ: Signature
∀ phi : Pattern,
  well_formed_closed (patt_exists phi) = true
  → wfc_body_ex phi
intros phi WFE.
Σ: Signature
phi: Pattern
WFE: well_formed_closed (patt_exists phi) = true
wfc_body_ex phi
unfold wfc_body_ex.
Σ: Signature
phi: Pattern
WFE: well_formed_closed (patt_exists phi) = true
∀ x : evar,
  x ∉ free_evars phi
  → well_formed_closed (evar_open 0 x phi) = true intros x H.
Σ: Signature
phi: Pattern
WFE: well_formed_closed (patt_exists phi) = true
x: evar
H: x ∉ free_evars phi
well_formed_closed (evar_open 0 x phi) = true
unfold well_formed_closed in *.
Σ: Signature
phi: Pattern
WFE: well_formed_closed_mu_aux (patt_exists phi) 0 &&
well_formed_closed_ex_aux (patt_exists phi) 0 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux (evar_open 0 x phi) 0 &&
well_formed_closed_ex_aux (evar_open 0 x phi) 0 = true simpl in WFE.
Σ: Signature
phi: Pattern
WFE: well_formed_closed_mu_aux phi 0 && well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux (evar_open 0 x phi) 0 &&
well_formed_closed_ex_aux (evar_open 0 x phi) 0 = true
apply andb_prop in WFE.
Σ: Signature
phi: Pattern
WFE: well_formed_closed_mu_aux phi 0 = true
∧ well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux (evar_open 0 x phi) 0 &&
well_formed_closed_ex_aux (evar_open 0 x phi) 0 = true destruct WFE as [WFE1 WFE2].
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux (evar_open 0 x phi) 0 &&
well_formed_closed_ex_aux (evar_open 0 x phi) 0 = true
rewrite wfc_ex_aux_body_ex_imp1.
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_ex_aux phi 1 = true
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux (evar_open 0 x phi) 0 &&
true = true auto.
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux (evar_open 0 x phi) 0 &&
true = true
rewrite wfc_mu_aux_body_ex_imp1.
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
well_formed_closed_mu_aux phi 0 = true
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
true && true = true auto.
Σ: Signature
phi: Pattern
WFE1: well_formed_closed_mu_aux phi 0 = true
WFE2: well_formed_closed_ex_aux phi 1 = true
x: evar
H: x ∉ free_evars phi
true && true = true
reflexivity.
Qed.



Lemma no_neg_occ_db_bevar_subst phi psi dbi1 dbi2:
well_formed_closed_mu_aux psi 0 = true ->
no_negative_occurrence_db_b dbi1 phi = true ->
no_negative_occurrence_db_b dbi1 (bevar_subst phi psi dbi2) = true
with no_pos_occ_db_bevar_subst  phi psi dbi1 dbi2:
   well_formed_closed_mu_aux psi 0 = true ->
   no_positive_occurrence_db_b dbi1 phi = true ->
   no_positive_occurrence_db_b dbi1 (bevar_subst phi psi dbi2) = true.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
dbi1, dbi2: db_index
well_formed_closed_mu_aux psi 0 = true
→ no_negative_occurrence_db_b dbi1 phi = true
  → no_negative_occurrence_db_b dbi1
      phi^[evar:dbi2↦psi] = true
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
dbi1, dbi2: db_index
well_formed_closed_mu_aux psi 0 = true
→ no_positive_occurrence_db_b dbi1 phi = true
  → no_positive_occurrence_db_b dbi1
      phi^[evar:dbi2↦psi] = true
Proof.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
dbi1, dbi2: db_index
well_formed_closed_mu_aux psi 0 = true
→ no_negative_occurrence_db_b dbi1 phi = true
  → no_negative_occurrence_db_b dbi1
      phi^[evar:dbi2↦psi] = true
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
dbi1, dbi2: db_index
well_formed_closed_mu_aux psi 0 = true
→ no_positive_occurrence_db_b dbi1 phi = true
  → no_positive_occurrence_db_b dbi1
      phi^[evar:dbi2↦psi] = true
-
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
dbi1, dbi2: db_index
well_formed_closed_mu_aux psi 0 = true
→ no_negative_occurrence_db_b dbi1 phi = true
  → no_negative_occurrence_db_b dbi1
      phi^[evar:dbi2↦psi] = true move: dbi1 dbi2.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi = true
    → no_negative_occurrence_db_b dbi1
        phi^[evar:dbi2↦psi] = true
induction phi; intros dbi1 dbi2 Hwfcpsi Hnonegphi; cbn in *; auto.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
no_negative_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: no_negative_occurrence_db_b dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2 =
true
no_negative_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: (fix no_negative_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2 =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
+
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
no_negative_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true case_match; auto.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
e: n = dbi2
H: compare_nat n dbi2 = Nat_equal n dbi2 e
no_negative_occurrence_db_b dbi1 psi = true now apply wfc_impl_no_neg_occ.
+
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: no_negative_occurrence_db_b dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2 =
true
no_negative_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true destruct_and!.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: no_negative_occurrence_db_b dbi1 phi1 = true
H0: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
rewrite -> IHphi1, -> IHphi2; auto.
+
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: (fix no_negative_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2 =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true destruct_and!.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H0: no_negative_occurrence_db_b dbi1 phi2 = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
fold (no_positive_occurrence_db_b dbi1 (bevar_subst phi1 psi dbi2)).
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H0: no_negative_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
rewrite no_pos_occ_db_bevar_subst; auto.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi1 =
    true
    → no_negative_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_negative_occurrence_db_b dbi1 phi2 =
    true
    → no_negative_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H0: no_negative_occurrence_db_b dbi1 phi2 = true
true &&
no_negative_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
rewrite -> IHphi2; auto.
-
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
dbi1, dbi2: db_index
well_formed_closed_mu_aux psi 0 = true
→ no_positive_occurrence_db_b dbi1 phi = true
  → no_positive_occurrence_db_b dbi1
      phi^[evar:dbi2↦psi] = true move: dbi1 dbi2.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi, psi: Pattern
∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi = true
    → no_positive_occurrence_db_b dbi1
        phi^[evar:dbi2↦psi] = true
induction phi; intros dbi1 dbi2 Hwfcpsi Hnonegphi; cbn in *; auto.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
no_positive_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: no_positive_occurrence_db_b dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2 =
true
no_positive_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: (fix no_negative_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2 =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
+
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
no_positive_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true repeat case_match; auto.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
e: n = dbi2
H: compare_nat n dbi2 = Nat_equal n dbi2 e
no_positive_occurrence_db_b dbi1 psi = true
apply wfc_impl_no_pos_occ.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
n: db_index
psi: Pattern
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: true = true
e: n = dbi2
H: compare_nat n dbi2 = Nat_equal n dbi2 e
well_formed_closed_mu_aux psi 0 = true assumption.
+
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: no_positive_occurrence_db_b dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2 =
true
no_positive_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true destruct_and!.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: no_positive_occurrence_db_b dbi1 phi1 = true
H0: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
rewrite -> IHphi1, -> IHphi2; auto.
+
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
Hnonegphi: (fix no_negative_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) 
   (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2 =
true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true destruct_and!.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H0: no_positive_occurrence_db_b dbi1 phi2 = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
fold (no_negative_occurrence_db_b dbi1 (bevar_subst phi1 psi dbi2)).
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H0: no_positive_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1 phi1^[evar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
rewrite no_neg_occ_db_bevar_subst; auto.
Σ: Signature
no_neg_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_negative_occurrence_db_b
      dbi1 phi = true
    → no_negative_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
no_pos_occ_db_bevar_subst: ∀ (phi psi : Pattern) 
  (dbi1 dbi2 : db_index),
  well_formed_closed_mu_aux
    psi 0 = true
  → no_positive_occurrence_db_b
      dbi1 phi = true
    → no_positive_occurrence_db_b
        dbi1
        phi^[evar:dbi2↦psi] =
      true
phi1, phi2, psi: Pattern
IHphi1: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi1 =
    true
    → no_positive_occurrence_db_b dbi1
        phi1^[evar:dbi2↦psi] = true
IHphi2: ∀ dbi1 dbi2 : db_index,
  well_formed_closed_mu_aux psi 0 = true
  → no_positive_occurrence_db_b dbi1 phi2 =
    true
    → no_positive_occurrence_db_b dbi1
        phi2^[evar:dbi2↦psi] = true
dbi1, dbi2: db_index
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H0: no_positive_occurrence_db_b dbi1 phi2 = true
true &&
no_positive_occurrence_db_b dbi1 phi2^[evar:dbi2↦psi] =
true
rewrite -> IHphi2; auto.
Qed.

Lemma bevar_subst_positive_2 :
forall φ ψ n,
well_formed_closed_mu_aux ψ 0 = true ->
well_formed_positive φ = true ->
well_formed_positive ψ = true ->
well_formed_positive (bevar_subst φ ψ n) = true.
Σ: Signature
∀ (φ ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
induction φ; intros ψ n' H0 H1 H2; cbn in *; auto.
Σ: Signature
n: db_index
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: true = true
H2: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: well_formed_positive φ1 &&
well_formed_positive φ2 = true
H2: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: well_formed_positive φ1 &&
well_formed_positive φ2 = true
H2: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ = true
H2: well_formed_positive ψ = true
no_negative_occurrence_db_b 0 φ^[evar:n'↦ψ] &&
well_formed_positive φ^[evar:n'↦ψ] = true
*
Σ: Signature
n: db_index
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: true = true
H2: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true break_match_goal; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: well_formed_positive φ1 &&
well_formed_positive φ2 = true
H2: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true destruct_and!.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H: well_formed_positive φ1 = true
H3: well_formed_positive φ2 = true
H2: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true rewrite -> IHφ1, -> IHφ2; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: well_formed_positive φ1 &&
well_formed_positive φ2 = true
H2: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true destruct_and!.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  well_formed_closed_mu_aux ψ 0 = true
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H: well_formed_positive φ1 = true
H3: well_formed_positive φ2 = true
H2: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true rewrite -> IHφ1, -> IHφ2; auto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : db_index),
well_formed_closed_mu_aux ψ 0 = true
→ well_formed_positive φ = true
→ well_formed_positive ψ = true
→ well_formed_positive φ^[evar:n↦ψ] = true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H1: no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ = true
H2: well_formed_positive ψ = true
no_negative_occurrence_db_b 0 φ^[evar:n'↦ψ] &&
well_formed_positive φ^[evar:n'↦ψ] = true destruct_and!.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : db_index),
well_formed_closed_mu_aux ψ 0 = true
→ well_formed_positive φ = true
→ well_formed_positive ψ = true
→ well_formed_positive φ^[evar:n↦ψ] = true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H: no_negative_occurrence_db_b 0 φ = true
H3: well_formed_positive φ = true
H2: well_formed_positive ψ = true
no_negative_occurrence_db_b 0 φ^[evar:n'↦ψ] &&
well_formed_positive φ^[evar:n'↦ψ] = true
rewrite IHφ; auto.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : db_index),
well_formed_closed_mu_aux ψ 0 = true
→ well_formed_positive φ = true
→ well_formed_positive ψ = true
→ well_formed_positive φ^[evar:n↦ψ] = true
ψ: Pattern
n': db_index
H0: well_formed_closed_mu_aux ψ 0 = true
H: no_negative_occurrence_db_b 0 φ = true
H3: well_formed_positive φ = true
H2: well_formed_positive ψ = true
no_negative_occurrence_db_b 0 φ^[evar:n'↦ψ] && true =
true
rewrite no_neg_occ_db_bevar_subst; auto.
Qed.

Corollary wfp_evar_open : forall phi x n,
well_formed_positive phi = true ->
well_formed_positive (evar_open n x phi) = true.
Σ: Signature
∀ (phi : Pattern) (x : evar) (n : db_index),
  well_formed_positive phi = true
  → well_formed_positive (evar_open n x phi) = true
Proof.
Σ: Signature
∀ (phi : Pattern) (x : evar) (n : db_index),
  well_formed_positive phi = true
  → well_formed_positive (evar_open n x phi) = true
intros phi x n WF.
Σ: Signature
phi: Pattern
x: evar
n: db_index
WF: well_formed_positive phi = true
well_formed_positive (evar_open n x phi) = true apply bevar_subst_positive_2; auto.
Qed.

(* Additional lemmas: evar_open, svar_open, freshness, well_formedness, etc. *)

(* evar_open and evar_quantify are inverses *)
Lemma evar_open_evar_quantify x n phi:
well_formed_closed_ex_aux phi n ->
(evar_open n x (evar_quantify x n phi)) = phi.
Σ: Signature
x: evar
n: db_index
phi: Pattern
well_formed_closed_ex_aux phi n
→ evar_open n x (evar_quantify x n phi) = phi
Proof.
Σ: Signature
x: evar
n: db_index
phi: Pattern
well_formed_closed_ex_aux phi n
→ evar_open n x (evar_quantify x n phi) = phi
intros H.
Σ: Signature
x: evar
n: db_index
phi: Pattern
H: well_formed_closed_ex_aux phi n
evar_open n x (evar_quantify x n phi) = phi
(*apply wfc_wfc_ind in H.*)
move: n H.
Σ: Signature
x: evar
phi: Pattern
∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → evar_open n x (evar_quantify x n phi) = phi
induction phi; intros n' H; cbn; auto.
Σ: Signature
x, x0: evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x0) n'
(if decide (x = x0)
 then patt_bound_evar n'
 else patt_free_evar x0)^[evar:n'↦patt_free_evar x] =
patt_free_evar x0
Σ: Signature
x: evar
n, n': db_index
H: well_formed_closed_ex_aux (patt_bound_evar n) n'
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → evar_open n x (evar_quantify x n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → evar_open n x (evar_quantify x n phi2) =
    phi2
n': db_index
H: well_formed_closed_ex_aux (patt_app phi1 phi2) n'
patt_app
  (evar_quantify x n' phi1)^[evar:n'↦patt_free_evar x]
  (evar_quantify x n' phi2)^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → evar_open n x (evar_quantify x n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → evar_open n x (evar_quantify x n phi2) =
    phi2
n': db_index
H: well_formed_closed_ex_aux (patt_imp phi1 phi2) n'
patt_imp
  (evar_quantify x n' phi1)^[evar:n'↦
  patt_free_evar x]
  (evar_quantify x n' phi2)^[evar:n'↦
  patt_free_evar x] = patt_imp phi1 phi2
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → evar_open n x (evar_quantify x n phi) =
    phi
n': db_index
H: well_formed_closed_ex_aux (patt_exists phi) n'
patt_exists
  (evar_quantify x (S n') phi)^[evar:
  S n'↦patt_free_evar x] = 
patt_exists phi
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → evar_open n x (evar_quantify x n phi) =
    phi
n': db_index
H: well_formed_closed_ex_aux (patt_mu phi) n'
patt_mu
  (evar_quantify x n' phi)^[evar:n'↦
  patt_free_evar x] = patt_mu phi
-
Σ: Signature
x, x0: evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x0) n'
(if decide (x = x0)
 then patt_bound_evar n'
 else patt_free_evar x0)^[evar:n'↦patt_free_evar x] =
patt_free_evar x0 destruct (decide (x = x0)); subst; simpl.
Σ: Signature
x0: evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x0) n'
match compare_nat n' n' with
| Nat_less _ _ _ => patt_bound_evar n'
| Nat_equal _ _ _ => patt_free_evar x0
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n')
end = patt_free_evar x0
Σ: Signature
x, x0: evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x0) n'
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0
+
Σ: Signature
x0: evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x0) n'
match compare_nat n' n' with
| Nat_less _ _ _ => patt_bound_evar n'
| Nat_equal _ _ _ => patt_free_evar x0
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n')
end = patt_free_evar x0 break_match_goal; auto; lia.
+
Σ: Signature
x, x0: evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x0) n'
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0 reflexivity.
-
Σ: Signature
x: evar
n, n': db_index
H: well_formed_closed_ex_aux (patt_bound_evar n) n'
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n simpl in *.
Σ: Signature
x: evar
n, n': db_index
H: if decide (n < n') then true else false
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n repeat case_match; simpl; auto; try lia; congruence.
-
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → evar_open n x (evar_quantify x n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → evar_open n x (evar_quantify x n phi2) =
    phi2
n': db_index
H: well_formed_closed_ex_aux (patt_app phi1 phi2) n'
patt_app
  (evar_quantify x n' phi1)^[evar:n'↦patt_free_evar x]
  (evar_quantify x n' phi2)^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2 cbn in H.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → evar_open n x (evar_quantify x n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → evar_open n x (evar_quantify x n phi2) =
    phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n'
patt_app
  (evar_quantify x n' phi1)^[evar:n'↦patt_free_evar x]
  (evar_quantify x n' phi2)^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2 simpl.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → evar_open n x (evar_quantify x n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → evar_open n x (evar_quantify x n phi2) =
    phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n'
patt_app
  (evar_quantify x n' phi1)^[evar:n'↦patt_free_evar x]
  (evar_quantify x n' phi2)^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2 unfold evar_open, evar_quantify in IHphi1, IHphi2.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n'
patt_app
  (evar_quantify x n' phi1)^[evar:n'↦patt_free_evar x]
  (evar_quantify x n' phi2)^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2
apply andb_true_iff in H.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' = true
∧ well_formed_closed_ex_aux phi2 n' = true
patt_app
  (evar_quantify x n' phi1)^[evar:n'↦patt_free_evar x]
  (evar_quantify x n' phi2)^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2
destruct H as [H1 H2].
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H1: well_formed_closed_ex_aux phi1 n' = true
H2: well_formed_closed_ex_aux phi2 n' = true
patt_app
  (evar_quantify x n' phi1)^[evar:n'↦patt_free_evar x]
  (evar_quantify x n' phi2)^[evar:n'↦patt_free_evar x] =
patt_app phi1 phi2
erewrite -> IHphi1, IHphi2 by eassumption.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H1: well_formed_closed_ex_aux phi1 n' = true
H2: well_formed_closed_ex_aux phi2 n' = true
patt_app phi1 phi2 = patt_app phi1 phi2
reflexivity.
-
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → evar_open n x (evar_quantify x n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → evar_open n x (evar_quantify x n phi2) =
    phi2
n': db_index
H: well_formed_closed_ex_aux (patt_imp phi1 phi2) n'
patt_imp
  (evar_quantify x n' phi1)^[evar:n'↦patt_free_evar x]
  (evar_quantify x n' phi2)^[evar:n'↦patt_free_evar x] =
patt_imp phi1 phi2 simpl in H.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → evar_open n x (evar_quantify x n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → evar_open n x (evar_quantify x n phi2) =
    phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n'
patt_imp
  (evar_quantify x n' phi1)^[evar:n'↦patt_free_evar x]
  (evar_quantify x n' phi2)^[evar:n'↦patt_free_evar x] =
patt_imp phi1 phi2 unfold evar_open, evar_quantify in IHphi1, IHphi2.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' &&
well_formed_closed_ex_aux phi2 n'
patt_imp
  (evar_quantify x n' phi1)^[evar:n'↦patt_free_evar x]
  (evar_quantify x n' phi2)^[evar:n'↦patt_free_evar x] =
patt_imp phi1 phi2
apply andb_true_iff in H.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H: well_formed_closed_ex_aux phi1 n' = true
∧ well_formed_closed_ex_aux phi2 n' = true
patt_imp
  (evar_quantify x n' phi1)^[evar:n'↦patt_free_evar x]
  (evar_quantify x n' phi2)^[evar:n'↦patt_free_evar x] =
patt_imp phi1 phi2
destruct H as [H1 H2].
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H1: well_formed_closed_ex_aux phi1 n' = true
H2: well_formed_closed_ex_aux phi2 n' = true
patt_imp
  (evar_quantify x n' phi1)^[evar:n'↦patt_free_evar x]
  (evar_quantify x n' phi2)^[evar:n'↦patt_free_evar x] =
patt_imp phi1 phi2
erewrite -> IHphi1, IHphi2 by eassumption.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_ex_aux phi1 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi1)^[evar:n↦
    patt_free_evar x] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_ex_aux phi2 n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu
              (evar_quantify x level p')
        end) x n phi2)^[evar:n↦
    patt_free_evar x] = phi2
n': db_index
H1: well_formed_closed_ex_aux phi1 n' = true
H2: well_formed_closed_ex_aux phi2 n' = true
patt_imp phi1 phi2 = patt_imp phi1 phi2
reflexivity.
-
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → evar_open n x (evar_quantify x n phi) =
    phi
n': db_index
H: well_formed_closed_ex_aux (patt_exists phi) n'
patt_exists
  (evar_quantify x (S n') phi)^[evar:S n'↦patt_free_evar
                                            x] =
patt_exists phi simpl in H.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → evar_open n x (evar_quantify x n phi) =
    phi
n': db_index
H: well_formed_closed_ex_aux phi (S n')
patt_exists
  (evar_quantify x (S n') phi)^[evar:S n'↦patt_free_evar
                                            x] =
patt_exists phi unfold evar_open, evar_quantify in IHphi.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu (evar_quantify x level p')
        end) x n phi)^[evar:n↦
    patt_free_evar x] = phi
n': db_index
H: well_formed_closed_ex_aux phi (S n')
patt_exists
  (evar_quantify x (S n') phi)^[evar:S n'↦patt_free_evar
                                            x] =
patt_exists phi
erewrite -> IHphi by eassumption.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → ((fix evar_quantify
        (x : evar) (level : db_index)
        (p : Pattern) {struct p} : Pattern :=
        match p with
        | patt_free_evar x' =>
            if decide (x = x')
            then patt_bound_evar level
            else patt_free_evar x'
        | patt_free_svar x' =>
            patt_free_svar x'
        | patt_bound_evar x' =>
            patt_bound_evar x'
        | patt_bound_svar X =>
            patt_bound_svar X
        | patt_sym s => patt_sym s
        | patt_app ls rs =>
            patt_app
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_bott => patt_bott
        | patt_imp ls rs =>
            patt_imp
              (evar_quantify x level ls)
              (evar_quantify x level rs)
        | patt_exists p' =>
            patt_exists
              (evar_quantify x (S level) p')
        | patt_mu p' =>
            patt_mu (evar_quantify x level p')
        end) x n phi)^[evar:n↦
    patt_free_evar x] = phi
n': db_index
H: well_formed_closed_ex_aux phi (S n')
patt_exists phi = patt_exists phi reflexivity.
-
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → evar_open n x (evar_quantify x n phi) =
    phi
n': db_index
H: well_formed_closed_ex_aux (patt_mu phi) n'
patt_mu
  (evar_quantify x n' phi)^[evar:n'↦patt_free_evar x] =
patt_mu phi simpl in H.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → evar_open n x (evar_quantify x n phi) =
    phi
n': db_index
H: well_formed_closed_ex_aux phi n'
patt_mu
  (evar_quantify x n' phi)^[evar:n'↦patt_free_evar x] =
patt_mu phi apply IHphi in H.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → evar_open n x (evar_quantify x n phi) =
    phi
n': db_index
H: evar_open n' x (evar_quantify x n' phi) = phi
patt_mu
  (evar_quantify x n' phi)^[evar:n'↦patt_free_evar x] =
patt_mu phi unfold evar_open in H.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → evar_open n x (evar_quantify x n phi) =
    phi
n': db_index
H: (evar_quantify x n' phi)^[evar:n'↦
patt_free_evar x] = phi
patt_mu
  (evar_quantify x n' phi)^[evar:n'↦patt_free_evar x] =
patt_mu phi rewrite H.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_ex_aux phi n
  → evar_open n x (evar_quantify x n phi) =
    phi
n': db_index
H: (evar_quantify x n' phi)^[evar:n'↦
patt_free_evar x] = phi
patt_mu phi = patt_mu phi reflexivity.
Qed.

Lemma svar_open_svar_quantify X n phi:
well_formed_closed_mu_aux phi n ->
(svar_open n X (svar_quantify X n phi)) = phi.
Σ: Signature
X: svar
n: db_index
phi: Pattern
well_formed_closed_mu_aux phi n
→ svar_open n X (svar_quantify X n phi) = phi
Proof.
Σ: Signature
X: svar
n: db_index
phi: Pattern
well_formed_closed_mu_aux phi n
→ svar_open n X (svar_quantify X n phi) = phi
intros H.
Σ: Signature
X: svar
n: db_index
phi: Pattern
H: well_formed_closed_mu_aux phi n
svar_open n X (svar_quantify X n phi) = phi
(*apply wfc_wfc_ind in H.*)
move: n H.
Σ: Signature
X: svar
phi: Pattern
∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → svar_open n X (svar_quantify X n phi) = phi
induction phi; intros n' H; cbn; auto.
Σ: Signature
X, x: svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
(if decide (X = x)
 then patt_bound_svar n'
 else patt_free_svar x)^[svar:n'↦patt_free_svar X] =
patt_free_svar x
Σ: Signature
X: svar
n, n': db_index
H: well_formed_closed_mu_aux (patt_bound_svar n) n'
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar X
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → svar_open n X (svar_quantify X n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → svar_open n X (svar_quantify X n phi2) =
    phi2
n': db_index
H: well_formed_closed_mu_aux (patt_app phi1 phi2) n'
patt_app
  (svar_quantify X n' phi1)^[svar:n'↦patt_free_svar X]
  (svar_quantify X n' phi2)^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → svar_open n X (svar_quantify X n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → svar_open n X (svar_quantify X n phi2) =
    phi2
n': db_index
H: well_formed_closed_mu_aux (patt_imp phi1 phi2) n'
patt_imp
  (svar_quantify X n' phi1)^[svar:n'↦
  patt_free_svar X]
  (svar_quantify X n' phi2)^[svar:n'↦
  patt_free_svar X] = patt_imp phi1 phi2
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → svar_open n X (svar_quantify X n phi) =
    phi
n': db_index
H: well_formed_closed_mu_aux (patt_exists phi) n'
patt_exists
  (svar_quantify X n' phi)^[svar:n'↦
  patt_free_svar X] = patt_exists phi
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → svar_open n X (svar_quantify X n phi) =
    phi
n': db_index
H: well_formed_closed_mu_aux (patt_mu phi) n'
patt_mu
  (svar_quantify X (S n') phi)^[svar:
  S n'↦patt_free_svar X] = 
patt_mu phi
-
Σ: Signature
X, x: svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
(if decide (X = x)
 then patt_bound_svar n'
 else patt_free_svar x)^[svar:n'↦patt_free_svar X] =
patt_free_svar x destruct (decide (X = x)); subst; simpl.
Σ: Signature
x: svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
match compare_nat n' n' with
| Nat_less _ _ _ => patt_bound_svar n'
| Nat_equal _ _ _ => patt_free_svar x
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n')
end = patt_free_svar x
Σ: Signature
X, x: svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
n: X ≠ x
patt_free_svar x = patt_free_svar x
+
Σ: Signature
x: svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
match compare_nat n' n' with
| Nat_less _ _ _ => patt_bound_svar n'
| Nat_equal _ _ _ => patt_free_svar x
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n')
end = patt_free_svar x break_match_goal; auto; lia.
+
Σ: Signature
X, x: svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
n: X ≠ x
patt_free_svar x = patt_free_svar x reflexivity.
-
Σ: Signature
X: svar
n, n': db_index
H: well_formed_closed_mu_aux (patt_bound_svar n) n'
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar X
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n simpl in *.
Σ: Signature
X: svar
n, n': db_index
H: if decide (n < n') then true else false
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar X
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n repeat case_match; simpl; auto; subst; try lia; try congruence.
-
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → svar_open n X (svar_quantify X n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → svar_open n X (svar_quantify X n phi2) =
    phi2
n': db_index
H: well_formed_closed_mu_aux (patt_app phi1 phi2) n'
patt_app
  (svar_quantify X n' phi1)^[svar:n'↦patt_free_svar X]
  (svar_quantify X n' phi2)^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2 cbn in H.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → svar_open n X (svar_quantify X n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → svar_open n X (svar_quantify X n phi2) =
    phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n'
patt_app
  (svar_quantify X n' phi1)^[svar:n'↦patt_free_svar X]
  (svar_quantify X n' phi2)^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2 simpl.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → svar_open n X (svar_quantify X n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → svar_open n X (svar_quantify X n phi2) =
    phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n'
patt_app
  (svar_quantify X n' phi1)^[svar:n'↦patt_free_svar X]
  (svar_quantify X n' phi2)^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2 unfold svar_open in IHphi1, IHphi2.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → (svar_quantify X n phi1)^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → (svar_quantify X n phi2)^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n'
patt_app
  (svar_quantify X n' phi1)^[svar:n'↦patt_free_svar X]
  (svar_quantify X n' phi2)^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2
apply andb_true_iff in H.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → (svar_quantify X n phi1)^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → (svar_quantify X n phi2)^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' = true
∧ well_formed_closed_mu_aux phi2 n' = true
patt_app
  (svar_quantify X n' phi1)^[svar:n'↦patt_free_svar X]
  (svar_quantify X n' phi2)^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2
destruct H as [H1 H2].
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → (svar_quantify X n phi1)^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → (svar_quantify X n phi2)^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H1: well_formed_closed_mu_aux phi1 n' = true
H2: well_formed_closed_mu_aux phi2 n' = true
patt_app
  (svar_quantify X n' phi1)^[svar:n'↦patt_free_svar X]
  (svar_quantify X n' phi2)^[svar:n'↦patt_free_svar X] =
patt_app phi1 phi2
erewrite -> IHphi1, IHphi2 by eassumption.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → (svar_quantify X n phi1)^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → (svar_quantify X n phi2)^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H1: well_formed_closed_mu_aux phi1 n' = true
H2: well_formed_closed_mu_aux phi2 n' = true
patt_app phi1 phi2 = patt_app phi1 phi2
reflexivity.
-
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → svar_open n X (svar_quantify X n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → svar_open n X (svar_quantify X n phi2) =
    phi2
n': db_index
H: well_formed_closed_mu_aux (patt_imp phi1 phi2) n'
patt_imp
  (svar_quantify X n' phi1)^[svar:n'↦patt_free_svar X]
  (svar_quantify X n' phi2)^[svar:n'↦patt_free_svar X] =
patt_imp phi1 phi2 simpl in H.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → svar_open n X (svar_quantify X n phi1) =
    phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → svar_open n X (svar_quantify X n phi2) =
    phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n'
patt_imp
  (svar_quantify X n' phi1)^[svar:n'↦patt_free_svar X]
  (svar_quantify X n' phi2)^[svar:n'↦patt_free_svar X] =
patt_imp phi1 phi2 unfold svar_open in IHphi1, IHphi2.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → (svar_quantify X n phi1)^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → (svar_quantify X n phi2)^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' &&
well_formed_closed_mu_aux phi2 n'
patt_imp
  (svar_quantify X n' phi1)^[svar:n'↦patt_free_svar X]
  (svar_quantify X n' phi2)^[svar:n'↦patt_free_svar X] =
patt_imp phi1 phi2
apply andb_true_iff in H.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → (svar_quantify X n phi1)^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → (svar_quantify X n phi2)^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H: well_formed_closed_mu_aux phi1 n' = true
∧ well_formed_closed_mu_aux phi2 n' = true
patt_imp
  (svar_quantify X n' phi1)^[svar:n'↦patt_free_svar X]
  (svar_quantify X n' phi2)^[svar:n'↦patt_free_svar X] =
patt_imp phi1 phi2
destruct H as [H1 H2].
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → (svar_quantify X n phi1)^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → (svar_quantify X n phi2)^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H1: well_formed_closed_mu_aux phi1 n' = true
H2: well_formed_closed_mu_aux phi2 n' = true
patt_imp
  (svar_quantify X n' phi1)^[svar:n'↦patt_free_svar X]
  (svar_quantify X n' phi2)^[svar:n'↦patt_free_svar X] =
patt_imp phi1 phi2
erewrite -> IHphi1, IHphi2 by eassumption.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : db_index,
  well_formed_closed_mu_aux phi1 n
  → (svar_quantify X n phi1)^[svar:n↦
    patt_free_svar X] = phi1
IHphi2: ∀ n : db_index,
  well_formed_closed_mu_aux phi2 n
  → (svar_quantify X n phi2)^[svar:n↦
    patt_free_svar X] = phi2
n': db_index
H1: well_formed_closed_mu_aux phi1 n' = true
H2: well_formed_closed_mu_aux phi2 n' = true
patt_imp phi1 phi2 = patt_imp phi1 phi2
reflexivity.
-
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → svar_open n X (svar_quantify X n phi) =
    phi
n': db_index
H: well_formed_closed_mu_aux (patt_exists phi) n'
patt_exists
  (svar_quantify X n' phi)^[svar:n'↦patt_free_svar X] =
patt_exists phi simpl in H.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → svar_open n X (svar_quantify X n phi) =
    phi
n': db_index
H: well_formed_closed_mu_aux phi n'
patt_exists
  (svar_quantify X n' phi)^[svar:n'↦patt_free_svar X] =
patt_exists phi unfold svar_open in IHphi.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → (svar_quantify X n phi)^[svar:n↦
    patt_free_svar X] = phi
n': db_index
H: well_formed_closed_mu_aux phi n'
patt_exists
  (svar_quantify X n' phi)^[svar:n'↦patt_free_svar X] =
patt_exists phi
erewrite -> IHphi by eassumption.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → (svar_quantify X n phi)^[svar:n↦
    patt_free_svar X] = phi
n': db_index
H: well_formed_closed_mu_aux phi n'
patt_exists phi = patt_exists phi reflexivity.
-
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → svar_open n X (svar_quantify X n phi) =
    phi
n': db_index
H: well_formed_closed_mu_aux (patt_mu phi) n'
patt_mu
  (svar_quantify X (S n') phi)^[svar:S n'↦patt_free_svar
                                            X] =
patt_mu phi simpl in H.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → svar_open n X (svar_quantify X n phi) =
    phi
n': db_index
H: well_formed_closed_mu_aux phi (S n')
patt_mu
  (svar_quantify X (S n') phi)^[svar:S n'↦patt_free_svar
                                            X] =
patt_mu phi apply IHphi in H.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → svar_open n X (svar_quantify X n phi) =
    phi
n': db_index
H: svar_open (S n') X (svar_quantify X (S n') phi) =
phi
patt_mu
  (svar_quantify X (S n') phi)^[svar:S n'↦patt_free_svar
                                            X] =
patt_mu phi unfold svar_open in H.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → svar_open n X (svar_quantify X n phi) =
    phi
n': db_index
H: (svar_quantify X (S n') phi)^[svar:
S n'↦patt_free_svar X] = phi
patt_mu
  (svar_quantify X (S n') phi)^[svar:S n'↦patt_free_svar
                                            X] =
patt_mu phi rewrite H.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : db_index,
  well_formed_closed_mu_aux phi n
  → svar_open n X (svar_quantify X n phi) =
    phi
n': db_index
H: (svar_quantify X (S n') phi)^[svar:
S n'↦patt_free_svar X] = phi
patt_mu phi = patt_mu phi reflexivity.
Qed.

Lemma evar_quantify_evar_open x n phi:
x ∉ free_evars phi -> well_formed_closed_ex_aux phi (S n) ->
(evar_quantify x n (evar_open n x phi)) = phi.
Σ: Signature
x: evar
n: nat
phi: Pattern
x ∉ free_evars phi
→ well_formed_closed_ex_aux phi (S n)
  → evar_quantify x n (evar_open n x phi) = phi
Proof.
Σ: Signature
x: evar
n: nat
phi: Pattern
x ∉ free_evars phi
→ well_formed_closed_ex_aux phi (S n)
  → evar_quantify x n (evar_open n x phi) = phi
revert n.
Σ: Signature
x: evar
phi: Pattern
∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → evar_quantify x n (evar_open n x phi) = phi
induction phi; intros n' H0 H1; simpl; auto.
Σ: Signature
x, x0: evar
n': nat
H0: x ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
(if decide (x = x0)
 then patt_bound_evar n'
 else patt_free_evar x0) = patt_free_evar x0
Σ: Signature
x: evar
n: db_index
n': nat
H0: x ∉ free_evars (patt_bound_evar n)
H1: well_formed_closed_ex_aux (patt_bound_evar n)
  (S n')
evar_quantify x n'
  (evar_open n' x (patt_bound_evar n)) =
patt_bound_evar n
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n (evar_open n x phi1) =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n (evar_open n x phi2) =
      phi2
n': nat
H0: x ∉ free_evars (patt_app phi1 phi2)
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
patt_app
  (evar_quantify x n' phi1^[evar:n'↦patt_free_evar x])
  (evar_quantify x n' phi2^[evar:n'↦patt_free_evar x]) =
patt_app phi1 phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n (evar_open n x phi1) =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n (evar_open n x phi2) =
      phi2
n': nat
H0: x ∉ free_evars (patt_imp phi1 phi2)
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
patt_imp
  (evar_quantify x n' phi1^[evar:n'↦patt_free_evar x])
  (evar_quantify x n' phi2^[evar:n'↦patt_free_evar x]) =
patt_imp phi1 phi2
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → evar_quantify x n (evar_open n x phi) =
      phi
n': nat
H0: x ∉ free_evars (patt_exists phi)
H1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n')
patt_exists
  (evar_quantify x (S n')
     phi^[evar:S n'↦patt_free_evar x]) =
patt_exists phi
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → evar_quantify x n (evar_open n x phi) =
      phi
n': nat
H0: x ∉ free_evars (patt_mu phi)
H1: well_formed_closed_ex_aux (patt_mu phi) (S n')
patt_mu
  (evar_quantify x n' phi^[evar:n'↦patt_free_evar x]) =
patt_mu phi
-
Σ: Signature
x, x0: evar
n': nat
H0: x ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
(if decide (x = x0)
 then patt_bound_evar n'
 else patt_free_evar x0) = patt_free_evar x0 destruct (decide (x = x0)); simpl.
Σ: Signature
x, x0: evar
n': nat
H0: x ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
e: x = x0
patt_bound_evar n' = patt_free_evar x0
Σ: Signature
x, x0: evar
n': nat
H0: x ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0
+
Σ: Signature
x, x0: evar
n': nat
H0: x ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
e: x = x0
patt_bound_evar n' = patt_free_evar x0 subst.
Σ: Signature
x0: evar
n': nat
H0: x0 ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
patt_bound_evar n' = patt_free_evar x0 simpl in H0.
Σ: Signature
x0: evar
n': nat
H0: x0 ∉ {[x0]}
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
patt_bound_evar n' = patt_free_evar x0 apply sets.not_elem_of_singleton_1 in H0.
Σ: Signature
x0: evar
n': nat
H0: x0 ≠ x0
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
patt_bound_evar n' = patt_free_evar x0 congruence.
+
Σ: Signature
x, x0: evar
n': nat
H0: x ∉ free_evars (patt_free_evar x0)
H1: well_formed_closed_ex_aux (patt_free_evar x0)
  (S n')
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0 reflexivity.
-
Σ: Signature
x: evar
n: db_index
n': nat
H0: x ∉ free_evars (patt_bound_evar n)
H1: well_formed_closed_ex_aux (patt_bound_evar n)
  (S n')
evar_quantify x n'
  (evar_open n' x (patt_bound_evar n)) =
patt_bound_evar n simpl in *.
Σ: Signature
x: evar
n: db_index
n': nat
H0: x ∉ ∅
H1: if decide (n < S n') then true else false
evar_quantify x n'
  (evar_open n' x (patt_bound_evar n)) =
patt_bound_evar n unfold evar_quantify,evar_open.
Σ: Signature
x: evar
n: db_index
n': nat
H0: x ∉ ∅
H1: if decide (n < S n') then true else false
(fix evar_quantify
   (x : evar) (level : db_index) (p : Pattern) {struct
    p} : Pattern :=
   match p with
   | patt_free_evar x' =>
       if decide (x = x')
       then patt_bound_evar level
       else patt_free_evar x'
   | patt_free_svar x' => patt_free_svar x'
   | patt_bound_evar x' => patt_bound_evar x'
   | patt_bound_svar X => patt_bound_svar X
   | patt_sym s => patt_sym s
   | patt_app ls rs =>
       patt_app (evar_quantify x level ls)
         (evar_quantify x level rs)
   | patt_bott => patt_bott
   | patt_imp ls rs =>
       patt_imp (evar_quantify x level ls)
         (evar_quantify x level rs)
   | patt_exists p' =>
       patt_exists (evar_quantify x (S level) p')
   | patt_mu p' => patt_mu (evar_quantify x level p')
   end) x n'
  (patt_bound_evar n)^[evar:n'↦patt_free_evar x] =
patt_bound_evar n simpl.
Σ: Signature
x: evar
n: db_index
n': nat
H0: x ∉ ∅
H1: if decide (n < S n') then true else false
(fix evar_quantify
   (x : evar) (level : db_index) (p : Pattern) {struct
    p} : Pattern :=
   match p with
   | patt_free_evar x' =>
       if decide (x = x')
       then patt_bound_evar level
       else patt_free_evar x'
   | patt_free_svar x' => patt_free_svar x'
   | patt_bound_evar x' => patt_bound_evar x'
   | patt_bound_svar X => patt_bound_svar X
   | patt_sym s => patt_sym s
   | patt_app ls rs =>
       patt_app (evar_quantify x level ls)
         (evar_quantify x level rs)
   | patt_bott => patt_bott
   | patt_imp ls rs =>
       patt_imp (evar_quantify x level ls)
         (evar_quantify x level rs)
   | patt_exists p' =>
       patt_exists (evar_quantify x (S level) p')
   | patt_mu p' => patt_mu (evar_quantify x level p')
   end) x n'
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = patt_bound_evar n
repeat case_match; auto; try congruence.
Σ: Signature
x: evar
n: db_index
n': nat
H0: x ∉ ∅
l: n < S n'
H: decide (n < S n') = left l
H1: true
g: n > n'
H2: compare_nat n n' = Nat_greater n n' g
patt_bound_evar (Nat.pred n) = patt_bound_evar n lia.
-
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n (evar_open n x phi1) =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n (evar_open n x phi2) =
      phi2
n': nat
H0: x ∉ free_evars (patt_app phi1 phi2)
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
patt_app
  (evar_quantify x n' phi1^[evar:n'↦patt_free_evar x])
  (evar_quantify x n' phi2^[evar:n'↦patt_free_evar x]) =
patt_app phi1 phi2 unfold evar_open in IHphi1, IHphi2.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H0: x ∉ free_evars (patt_app phi1 phi2)
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
patt_app
  (evar_quantify x n' phi1^[evar:n'↦patt_free_evar x])
  (evar_quantify x n' phi2^[evar:n'↦patt_free_evar x]) =
patt_app phi1 phi2
rewrite sets.not_elem_of_union in H0.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
H0: (x
 ∉ (fix free_evars (phi : Pattern) : EVarSet :=
      match phi with
      | patt_free_evar x => {[x]}
      | patt_app phi1 phi2 |
        patt_imp phi1 phi2 =>
          free_evars phi1 ∪ free_evars phi2
      | patt_exists phi0 | patt_mu phi0 =>
          free_evars phi0
      | _ => ∅
      end) phi1)
∧ x
  ∉ (fix free_evars (phi : Pattern) : EVarSet :=
       match phi with
       | patt_free_evar x => {[x]}
       | patt_app phi1 phi2 |
         patt_imp phi1 phi2 =>
           free_evars phi1 ∪ free_evars phi2
       | patt_exists phi0 | patt_mu phi0 =>
           free_evars phi0
       | _ => ∅
       end) phi2
patt_app
  (evar_quantify x n' phi1^[evar:n'↦patt_free_evar x])
  (evar_quantify x n' phi2^[evar:n'↦patt_free_evar x]) =
patt_app phi1 phi2 destruct H0 as [E1 E2].
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
patt_app
  (evar_quantify x n' phi1^[evar:n'↦patt_free_evar x])
  (evar_quantify x n' phi2^[evar:n'↦patt_free_evar x]) =
patt_app phi1 phi2
rewrite -> IHphi1, IHphi2.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
patt_app phi1 phi2 = patt_app phi1 phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi2 (S n')
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi1
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi1 (S n')
reflexivity.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi2 (S n')
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi1
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_app phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi1 (S n')
all: auto; apply andb_true_iff in H1; apply H1.
-
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n (evar_open n x phi1) =
      phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n (evar_open n x phi2) =
      phi2
n': nat
H0: x ∉ free_evars (patt_imp phi1 phi2)
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
patt_imp
  (evar_quantify x n' phi1^[evar:n'↦patt_free_evar x])
  (evar_quantify x n' phi2^[evar:n'↦patt_free_evar x]) =
patt_imp phi1 phi2 unfold evar_open in IHphi1, IHphi2.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H0: x ∉ free_evars (patt_imp phi1 phi2)
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
patt_imp
  (evar_quantify x n' phi1^[evar:n'↦patt_free_evar x])
  (evar_quantify x n' phi2^[evar:n'↦patt_free_evar x]) =
patt_imp phi1 phi2
rewrite sets.not_elem_of_union in H0.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
H0: (x
 ∉ (fix free_evars (phi : Pattern) : EVarSet :=
      match phi with
      | patt_free_evar x => {[x]}
      | patt_app phi1 phi2 |
        patt_imp phi1 phi2 =>
          free_evars phi1 ∪ free_evars phi2
      | patt_exists phi0 | patt_mu phi0 =>
          free_evars phi0
      | _ => ∅
      end) phi1)
∧ x
  ∉ (fix free_evars (phi : Pattern) : EVarSet :=
       match phi with
       | patt_free_evar x => {[x]}
       | patt_app phi1 phi2 |
         patt_imp phi1 phi2 =>
           free_evars phi1 ∪ free_evars phi2
       | patt_exists phi0 | patt_mu phi0 =>
           free_evars phi0
       | _ => ∅
       end) phi2
patt_imp
  (evar_quantify x n' phi1^[evar:n'↦patt_free_evar x])
  (evar_quantify x n' phi2^[evar:n'↦patt_free_evar x]) =
patt_imp phi1 phi2 destruct H0 as [E1 E2].
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
patt_imp
  (evar_quantify x n' phi1^[evar:n'↦patt_free_evar x])
  (evar_quantify x n' phi2^[evar:n'↦patt_free_evar x]) =
patt_imp phi1 phi2
rewrite -> IHphi1, IHphi2.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
patt_imp phi1 phi2 = patt_imp phi1 phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi2 (S n')
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi1
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi1 (S n')
reflexivity.
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi2
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi2 (S n')
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
x ∉ free_evars phi1
Σ: Signature
x: evar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  x ∉ free_evars phi1
  → well_formed_closed_ex_aux phi1 (S n)
    → evar_quantify x n
        phi1^[evar:n↦patt_free_evar x] = phi1
IHphi2: ∀ n : nat,
  x ∉ free_evars phi2
  → well_formed_closed_ex_aux phi2 (S n)
    → evar_quantify x n
        phi2^[evar:n↦patt_free_evar x] = phi2
n': nat
H1: well_formed_closed_ex_aux 
  (patt_imp phi1 phi2) (S n')
E1: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi1
E2: x
∉ (fix free_evars (phi : Pattern) : EVarSet :=
     match phi with
     | patt_free_evar x => {[x]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_evars phi1 ∪ free_evars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_evars phi0
     | _ => ∅
     end) phi2
well_formed_closed_ex_aux phi1 (S n')
all: auto; apply andb_true_iff in H1; apply H1.
-
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → evar_quantify x n (evar_open n x phi) =
      phi
n': nat
H0: x ∉ free_evars (patt_exists phi)
H1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n')
patt_exists
  (evar_quantify x (S n')
     phi^[evar:S n'↦patt_free_evar x]) =
patt_exists phi simpl in H0.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → evar_quantify x n (evar_open n x phi) =
      phi
n': nat
H0: x ∉ free_evars phi
H1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n')
patt_exists
  (evar_quantify x (S n')
     phi^[evar:S n'↦patt_free_evar x]) =
patt_exists phi unfold evar_open in IHphi.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → evar_quantify x n
        phi^[evar:n↦patt_free_evar x] = phi
n': nat
H0: x ∉ free_evars phi
H1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n')
patt_exists
  (evar_quantify x (S n')
     phi^[evar:S n'↦patt_free_evar x]) =
patt_exists phi
rewrite -> IHphi by assumption.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → evar_quantify x n
        phi^[evar:n↦patt_free_evar x] = phi
n': nat
H0: x ∉ free_evars phi
H1: well_formed_closed_ex_aux 
  (patt_exists phi) (S n')
patt_exists phi = patt_exists phi reflexivity.
-
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → evar_quantify x n (evar_open n x phi) =
      phi
n': nat
H0: x ∉ free_evars (patt_mu phi)
H1: well_formed_closed_ex_aux (patt_mu phi) (S n')
patt_mu
  (evar_quantify x n' phi^[evar:n'↦patt_free_evar x]) =
patt_mu phi simpl in H0.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → evar_quantify x n (evar_open n x phi) =
      phi
n': nat
H0: x ∉ free_evars phi
H1: well_formed_closed_ex_aux (patt_mu phi) (S n')
patt_mu
  (evar_quantify x n' phi^[evar:n'↦patt_free_evar x]) =
patt_mu phi unfold evar_open in IHphi.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → evar_quantify x n
        phi^[evar:n↦patt_free_evar x] = phi
n': nat
H0: x ∉ free_evars phi
H1: well_formed_closed_ex_aux (patt_mu phi) (S n')
patt_mu
  (evar_quantify x n' phi^[evar:n'↦patt_free_evar x]) =
patt_mu phi
rewrite -> IHphi by assumption.
Σ: Signature
x: evar
phi: Pattern
IHphi: ∀ n : nat,
  x ∉ free_evars phi
  → well_formed_closed_ex_aux phi (S n)
    → evar_quantify x n
        phi^[evar:n↦patt_free_evar x] = phi
n': nat
H0: x ∉ free_evars phi
H1: well_formed_closed_ex_aux (patt_mu phi) (S n')
patt_mu phi = patt_mu phi reflexivity.
Qed.

Lemma svar_quantify_svar_open X n phi:
X ∉ free_svars phi -> well_formed_closed_mu_aux phi (S n) ->
(svar_quantify X n (svar_open n X phi)) = phi.
Σ: Signature
X: svar
n: nat
phi: Pattern
X ∉ free_svars phi
→ well_formed_closed_mu_aux phi (S n)
  → svar_quantify X n (svar_open n X phi) = phi
Proof.
Σ: Signature
X: svar
n: nat
phi: Pattern
X ∉ free_svars phi
→ well_formed_closed_mu_aux phi (S n)
  → svar_quantify X n (svar_open n X phi) = phi
revert n.
Σ: Signature
X: svar
phi: Pattern
∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → svar_quantify X n (svar_open n X phi) = phi
induction phi; intros n' H0 H1; simpl; auto.
Σ: Signature
X, x: svar
n': nat
H0: X ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
(if decide (X = x)
 then patt_bound_svar n'
 else patt_free_svar x) = patt_free_svar x
Σ: Signature
X: svar
n: db_index
n': nat
H0: X ∉ free_svars (patt_bound_svar n)
H1: well_formed_closed_mu_aux (patt_bound_svar n)
  (S n')
svar_quantify X n'
  (svar_open n' X (patt_bound_svar n)) =
patt_bound_svar n
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n (svar_open n X phi1) =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n (svar_open n X phi2) =
      phi2
n': nat
H0: X ∉ free_svars (patt_app phi1 phi2)
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
patt_app
  (svar_quantify X n' phi1^[svar:n'↦patt_free_svar X])
  (svar_quantify X n' phi2^[svar:n'↦patt_free_svar X]) =
patt_app phi1 phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n (svar_open n X phi1) =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n (svar_open n X phi2) =
      phi2
n': nat
H0: X ∉ free_svars (patt_imp phi1 phi2)
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
patt_imp
  (svar_quantify X n' phi1^[svar:n'↦patt_free_svar X])
  (svar_quantify X n' phi2^[svar:n'↦patt_free_svar X]) =
patt_imp phi1 phi2
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → svar_quantify X n (svar_open n X phi) =
      phi
n': nat
H0: X ∉ free_svars (patt_exists phi)
H1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n')
patt_exists
  (svar_quantify X n' phi^[svar:n'↦patt_free_svar X]) =
patt_exists phi
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → svar_quantify X n (svar_open n X phi) =
      phi
n': nat
H0: X ∉ free_svars (patt_mu phi)
H1: well_formed_closed_mu_aux (patt_mu phi) (S n')
patt_mu
  (svar_quantify X (S n')
     phi^[svar:S n'↦patt_free_svar X]) = 
patt_mu phi
-
Σ: Signature
X, x: svar
n': nat
H0: X ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
(if decide (X = x)
 then patt_bound_svar n'
 else patt_free_svar x) = patt_free_svar x destruct (decide (X = x)); simpl.
Σ: Signature
X, x: svar
n': nat
H0: X ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
e: X = x
patt_bound_svar n' = patt_free_svar x
Σ: Signature
X, x: svar
n': nat
H0: X ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
n: X ≠ x
patt_free_svar x = patt_free_svar x
+
Σ: Signature
X, x: svar
n': nat
H0: X ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
e: X = x
patt_bound_svar n' = patt_free_svar x subst.
Σ: Signature
x: svar
n': nat
H0: x ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
patt_bound_svar n' = patt_free_svar x simpl in H0.
Σ: Signature
x: svar
n': nat
H0: x ∉ {[x]}
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
patt_bound_svar n' = patt_free_svar x apply sets.not_elem_of_singleton_1 in H0.
Σ: Signature
x: svar
n': nat
H0: x ≠ x
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
patt_bound_svar n' = patt_free_svar x congruence.
+
Σ: Signature
X, x: svar
n': nat
H0: X ∉ free_svars (patt_free_svar x)
H1: well_formed_closed_mu_aux (patt_free_svar x)
  (S n')
n: X ≠ x
patt_free_svar x = patt_free_svar x reflexivity.
-
Σ: Signature
X: svar
n: db_index
n': nat
H0: X ∉ free_svars (patt_bound_svar n)
H1: well_formed_closed_mu_aux (patt_bound_svar n)
  (S n')
svar_quantify X n'
  (svar_open n' X (patt_bound_svar n)) =
patt_bound_svar n simpl in *.
Σ: Signature
X: svar
n: db_index
n': nat
H0: X ∉ ∅
H1: if decide (n < S n') then true else false
svar_quantify X n'
  (svar_open n' X (patt_bound_svar n)) =
patt_bound_svar n unfold svar_quantify,svar_open,bsvar_subst.
Σ: Signature
X: svar
n: db_index
n': nat
H0: X ∉ ∅
H1: if decide (n < S n') then true else false
(fix svar_quantify
   (X : svar) (level : db_index) (p : Pattern) {struct
    p} : Pattern :=
   match p with
   | patt_free_evar x' => patt_free_evar x'
   | patt_free_svar X' =>
       if decide (X = X')
       then patt_bound_svar level
       else patt_free_svar X'
   | patt_bound_evar x' => patt_bound_evar x'
   | patt_bound_svar X0 => patt_bound_svar X0
   | patt_sym s => patt_sym s
   | patt_app ls rs =>
       patt_app (svar_quantify X level ls)
         (svar_quantify X level rs)
   | patt_bott => patt_bott
   | patt_imp ls rs =>
       patt_imp (svar_quantify X level ls)
         (svar_quantify X level rs)
   | patt_exists p' =>
       patt_exists (svar_quantify X level p')
   | patt_mu p' =>
       patt_mu (svar_quantify X (S level) p')
   end) X n'
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = patt_bound_svar n
repeat case_match; auto; try congruence.
Σ: Signature
X: svar
n: db_index
n': nat
H0: X ∉ ∅
l: n < S n'
H: decide (n < S n') = left l
H1: true
g: n > n'
H2: compare_nat n n' = Nat_greater n n' g
patt_bound_svar (Nat.pred n) = patt_bound_svar n lia.
-
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n (svar_open n X phi1) =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n (svar_open n X phi2) =
      phi2
n': nat
H0: X ∉ free_svars (patt_app phi1 phi2)
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
patt_app
  (svar_quantify X n' phi1^[svar:n'↦patt_free_svar X])
  (svar_quantify X n' phi2^[svar:n'↦patt_free_svar X]) =
patt_app phi1 phi2 unfold svar_open in IHphi1, IHphi2.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H0: X ∉ free_svars (patt_app phi1 phi2)
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
patt_app
  (svar_quantify X n' phi1^[svar:n'↦patt_free_svar X])
  (svar_quantify X n' phi2^[svar:n'↦patt_free_svar X]) =
patt_app phi1 phi2
rewrite sets.not_elem_of_union in H0.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
H0: (X
 ∉ (fix free_svars (phi : Pattern) : SVarSet :=
      match phi with
      | patt_free_svar X => {[X]}
      | patt_app phi1 phi2 |
        patt_imp phi1 phi2 =>
          free_svars phi1 ∪ free_svars phi2
      | patt_exists phi0 | patt_mu phi0 =>
          free_svars phi0
      | _ => ∅
      end) phi1)
∧ X
  ∉ (fix free_svars (phi : Pattern) : SVarSet :=
       match phi with
       | patt_free_svar X => {[X]}
       | patt_app phi1 phi2 |
         patt_imp phi1 phi2 =>
           free_svars phi1 ∪ free_svars phi2
       | patt_exists phi0 | patt_mu phi0 =>
           free_svars phi0
       | _ => ∅
       end) phi2
patt_app
  (svar_quantify X n' phi1^[svar:n'↦patt_free_svar X])
  (svar_quantify X n' phi2^[svar:n'↦patt_free_svar X]) =
patt_app phi1 phi2 destruct H0 as [E1 E2].
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
patt_app
  (svar_quantify X n' phi1^[svar:n'↦patt_free_svar X])
  (svar_quantify X n' phi2^[svar:n'↦patt_free_svar X]) =
patt_app phi1 phi2
rewrite -> IHphi1, IHphi2.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
patt_app phi1 phi2 = patt_app phi1 phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi2 (S n')
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi1
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi1 (S n')
reflexivity.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi2 (S n')
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi1
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_app phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi1 (S n')
all: auto; apply andb_true_iff in H1; apply H1.
-
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n (svar_open n X phi1) =
      phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n (svar_open n X phi2) =
      phi2
n': nat
H0: X ∉ free_svars (patt_imp phi1 phi2)
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
patt_imp
  (svar_quantify X n' phi1^[svar:n'↦patt_free_svar X])
  (svar_quantify X n' phi2^[svar:n'↦patt_free_svar X]) =
patt_imp phi1 phi2 unfold svar_open in IHphi1, IHphi2.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H0: X ∉ free_svars (patt_imp phi1 phi2)
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
patt_imp
  (svar_quantify X n' phi1^[svar:n'↦patt_free_svar X])
  (svar_quantify X n' phi2^[svar:n'↦patt_free_svar X]) =
patt_imp phi1 phi2
rewrite sets.not_elem_of_union in H0.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
H0: (X
 ∉ (fix free_svars (phi : Pattern) : SVarSet :=
      match phi with
      | patt_free_svar X => {[X]}
      | patt_app phi1 phi2 |
        patt_imp phi1 phi2 =>
          free_svars phi1 ∪ free_svars phi2
      | patt_exists phi0 | patt_mu phi0 =>
          free_svars phi0
      | _ => ∅
      end) phi1)
∧ X
  ∉ (fix free_svars (phi : Pattern) : SVarSet :=
       match phi with
       | patt_free_svar X => {[X]}
       | patt_app phi1 phi2 |
         patt_imp phi1 phi2 =>
           free_svars phi1 ∪ free_svars phi2
       | patt_exists phi0 | patt_mu phi0 =>
           free_svars phi0
       | _ => ∅
       end) phi2
patt_imp
  (svar_quantify X n' phi1^[svar:n'↦patt_free_svar X])
  (svar_quantify X n' phi2^[svar:n'↦patt_free_svar X]) =
patt_imp phi1 phi2 destruct H0 as [E1 E2].
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
patt_imp
  (svar_quantify X n' phi1^[svar:n'↦patt_free_svar X])
  (svar_quantify X n' phi2^[svar:n'↦patt_free_svar X]) =
patt_imp phi1 phi2
rewrite -> IHphi1, IHphi2.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
patt_imp phi1 phi2 = patt_imp phi1 phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi2 (S n')
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi1
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi1 (S n')
reflexivity.
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi2
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi2 (S n')
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
X ∉ free_svars phi1
Σ: Signature
X: svar
phi1, phi2: Pattern
IHphi1: ∀ n : nat,
  X ∉ free_svars phi1
  → well_formed_closed_mu_aux phi1 (S n)
    → svar_quantify X n
        phi1^[svar:n↦patt_free_svar X] = phi1
IHphi2: ∀ n : nat,
  X ∉ free_svars phi2
  → well_formed_closed_mu_aux phi2 (S n)
    → svar_quantify X n
        phi2^[svar:n↦patt_free_svar X] = phi2
n': nat
H1: well_formed_closed_mu_aux 
  (patt_imp phi1 phi2) (S n')
E1: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi1
E2: X
∉ (fix free_svars (phi : Pattern) : SVarSet :=
     match phi with
     | patt_free_svar X => {[X]}
     | patt_app phi1 phi2 | patt_imp phi1 phi2 =>
         free_svars phi1 ∪ free_svars phi2
     | patt_exists phi0 | patt_mu phi0 =>
         free_svars phi0
     | _ => ∅
     end) phi2
well_formed_closed_mu_aux phi1 (S n')
all: auto; apply andb_true_iff in H1; apply H1.
-
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → svar_quantify X n (svar_open n X phi) =
      phi
n': nat
H0: X ∉ free_svars (patt_exists phi)
H1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n')
patt_exists
  (svar_quantify X n' phi^[svar:n'↦patt_free_svar X]) =
patt_exists phi simpl in H0.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → svar_quantify X n (svar_open n X phi) =
      phi
n': nat
H0: X ∉ free_svars phi
H1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n')
patt_exists
  (svar_quantify X n' phi^[svar:n'↦patt_free_svar X]) =
patt_exists phi unfold svar_open in IHphi.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → svar_quantify X n
        phi^[svar:n↦patt_free_svar X] = phi
n': nat
H0: X ∉ free_svars phi
H1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n')
patt_exists
  (svar_quantify X n' phi^[svar:n'↦patt_free_svar X]) =
patt_exists phi
erewrite -> IHphi by assumption.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → svar_quantify X n
        phi^[svar:n↦patt_free_svar X] = phi
n': nat
H0: X ∉ free_svars phi
H1: well_formed_closed_mu_aux 
  (patt_exists phi) (S n')
patt_exists phi = patt_exists phi reflexivity.
-
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → svar_quantify X n (svar_open n X phi) =
      phi
n': nat
H0: X ∉ free_svars (patt_mu phi)
H1: well_formed_closed_mu_aux (patt_mu phi) (S n')
patt_mu
  (svar_quantify X (S n')
     phi^[svar:S n'↦patt_free_svar X]) = patt_mu phi simpl in H0.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → svar_quantify X n (svar_open n X phi) =
      phi
n': nat
H0: X ∉ free_svars phi
H1: well_formed_closed_mu_aux (patt_mu phi) (S n')
patt_mu
  (svar_quantify X (S n')
     phi^[svar:S n'↦patt_free_svar X]) = patt_mu phi unfold svar_open in IHphi.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → svar_quantify X n
        phi^[svar:n↦patt_free_svar X] = phi
n': nat
H0: X ∉ free_svars phi
H1: well_formed_closed_mu_aux (patt_mu phi) (S n')
patt_mu
  (svar_quantify X (S n')
     phi^[svar:S n'↦patt_free_svar X]) = patt_mu phi
erewrite -> IHphi by assumption.
Σ: Signature
X: svar
phi: Pattern
IHphi: ∀ n : nat,
  X ∉ free_svars phi
  → well_formed_closed_mu_aux phi (S n)
    → svar_quantify X n
        phi^[svar:n↦patt_free_svar X] = phi
n': nat
H0: X ∉ free_svars phi
H1: well_formed_closed_mu_aux (patt_mu phi) (S n')
patt_mu phi = patt_mu phi reflexivity.
Qed.

Lemma double_evar_quantify φ : forall x n,
evar_quantify x n (evar_quantify x n φ) = evar_quantify x n φ.
Σ: Signature
φ: Pattern
∀ (x : evar) (n : db_index),
  evar_quantify x n (evar_quantify x n φ) =
  evar_quantify x n φ
Proof.
Σ: Signature
φ: Pattern
∀ (x : evar) (n : db_index),
  evar_quantify x n (evar_quantify x n φ) =
  evar_quantify x n φ
induction φ; intros x' n'; simpl; auto.
Σ: Signature
x, x': evar
n': db_index
evar_quantify x' n'
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) =
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  evar_quantify x n (evar_quantify x n φ1) =
  evar_quantify x n φ1
IHφ2: ∀ (x : evar) (n : db_index),
  evar_quantify x n (evar_quantify x n φ2) =
  evar_quantify x n φ2
x': evar
n': db_index
patt_app
  (evar_quantify x' n' (evar_quantify x' n' φ1))
  (evar_quantify x' n' (evar_quantify x' n' φ2)) =
patt_app (evar_quantify x' n' φ1)
  (evar_quantify x' n' φ2)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  evar_quantify x n (evar_quantify x n φ1) =
  evar_quantify x n φ1
IHφ2: ∀ (x : evar) (n : db_index),
  evar_quantify x n (evar_quantify x n φ2) =
  evar_quantify x n φ2
x': evar
n': db_index
patt_imp
  (evar_quantify x' n' (evar_quantify x' n' φ1))
  (evar_quantify x' n' (evar_quantify x' n' φ2)) =
patt_imp (evar_quantify x' n' φ1)
  (evar_quantify x' n' φ2)
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  evar_quantify x n (evar_quantify x n φ) =
  evar_quantify x n φ
x': evar
n': db_index
patt_exists
  (evar_quantify x' (S n') (evar_quantify x' (S n') φ)) =
patt_exists (evar_quantify x' (S n') φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  evar_quantify x n (evar_quantify x n φ) =
  evar_quantify x n φ
x': evar
n': db_index
patt_mu (evar_quantify x' n' (evar_quantify x' n' φ)) =
patt_mu (evar_quantify x' n' φ)
*
Σ: Signature
x, x': evar
n': db_index
evar_quantify x' n'
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) =
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) unfold evar_quantify.
Σ: Signature
x, x': evar
n': db_index
(fix evar_quantify
   (x : evar) (level : db_index) (p : Pattern) {struct
    p} : Pattern :=
   match p with
   | patt_free_evar x' =>
       if decide (x = x')
       then patt_bound_evar level
       else patt_free_evar x'
   | patt_free_svar x' => patt_free_svar x'
   | patt_bound_evar x' => patt_bound_evar x'
   | patt_bound_svar X => patt_bound_svar X
   | patt_sym s => patt_sym s
   | patt_app ls rs =>
       patt_app (evar_quantify x level ls)
         (evar_quantify x level rs)
   | patt_bott => patt_bott
   | patt_imp ls rs =>
       patt_imp (evar_quantify x level ls)
         (evar_quantify x level rs)
   | patt_exists p' =>
       patt_exists (evar_quantify x (S level) p')
   | patt_mu p' => patt_mu (evar_quantify x level p')
   end) x' n'
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) =
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) repeat case_match; auto.
Σ: Signature
x, x': evar
n': db_index
n: x' ≠ x
e: x' = x
H0: decide (x' = x) = left e
H: left e = right n
patt_bound_evar n' = patt_free_evar x contradiction.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  evar_quantify x n (evar_quantify x n φ1) =
  evar_quantify x n φ1
IHφ2: ∀ (x : evar) (n : db_index),
  evar_quantify x n (evar_quantify x n φ2) =
  evar_quantify x n φ2
x': evar
n': db_index
patt_app
  (evar_quantify x' n' (evar_quantify x' n' φ1))
  (evar_quantify x' n' (evar_quantify x' n' φ2)) =
patt_app (evar_quantify x' n' φ1)
  (evar_quantify x' n' φ2) now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  evar_quantify x n (evar_quantify x n φ1) =
  evar_quantify x n φ1
IHφ2: ∀ (x : evar) (n : db_index),
  evar_quantify x n (evar_quantify x n φ2) =
  evar_quantify x n φ2
x': evar
n': db_index
patt_imp
  (evar_quantify x' n' (evar_quantify x' n' φ1))
  (evar_quantify x' n' (evar_quantify x' n' φ2)) =
patt_imp (evar_quantify x' n' φ1)
  (evar_quantify x' n' φ2) now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
evar_quantify x n (evar_quantify x n φ) = evar_quantify x n φ
x': evar
n': db_index
patt_exists
  (evar_quantify x' (S n') (evar_quantify x' (S n') φ)) =
patt_exists (evar_quantify x' (S n') φ) now rewrite IHφ.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
evar_quantify x n (evar_quantify x n φ) = evar_quantify x n φ
x': evar
n': db_index
patt_mu (evar_quantify x' n' (evar_quantify x' n' φ)) =
patt_mu (evar_quantify x' n' φ) now rewrite IHφ.
Qed.

Lemma double_svar_quantify φ : forall X n,
svar_quantify X n (svar_quantify X n φ) = svar_quantify X n φ.
Σ: Signature
φ: Pattern
∀ (X : svar) (n : db_index),
  svar_quantify X n (svar_quantify X n φ) =
  svar_quantify X n φ
Proof.
Σ: Signature
φ: Pattern
∀ (X : svar) (n : db_index),
  svar_quantify X n (svar_quantify X n φ) =
  svar_quantify X n φ
induction φ; intros x' n'; simpl; auto.
Σ: Signature
x, x': svar
n': db_index
svar_quantify x' n'
  (if decide (x' = x)
   then patt_bound_svar n'
   else patt_free_svar x) =
(if decide (x' = x)
 then patt_bound_svar n'
 else patt_free_svar x)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  svar_quantify X n (svar_quantify X n φ1) =
  svar_quantify X n φ1
IHφ2: ∀ (X : svar) (n : db_index),
  svar_quantify X n (svar_quantify X n φ2) =
  svar_quantify X n φ2
x': svar
n': db_index
patt_app
  (svar_quantify x' n' (svar_quantify x' n' φ1))
  (svar_quantify x' n' (svar_quantify x' n' φ2)) =
patt_app (svar_quantify x' n' φ1)
  (svar_quantify x' n' φ2)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  svar_quantify X n (svar_quantify X n φ1) =
  svar_quantify X n φ1
IHφ2: ∀ (X : svar) (n : db_index),
  svar_quantify X n (svar_quantify X n φ2) =
  svar_quantify X n φ2
x': svar
n': db_index
patt_imp
  (svar_quantify x' n' (svar_quantify x' n' φ1))
  (svar_quantify x' n' (svar_quantify x' n' φ2)) =
patt_imp (svar_quantify x' n' φ1)
  (svar_quantify x' n' φ2)
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (n : db_index),
  svar_quantify X n (svar_quantify X n φ) =
  svar_quantify X n φ
x': svar
n': db_index
patt_exists
  (svar_quantify x' n' (svar_quantify x' n' φ)) =
patt_exists (svar_quantify x' n' φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (n : db_index),
  svar_quantify X n (svar_quantify X n φ) =
  svar_quantify X n φ
x': svar
n': db_index
patt_mu
  (svar_quantify x' (S n') (svar_quantify x' (S n') φ)) =
patt_mu (svar_quantify x' (S n') φ)
*
Σ: Signature
x, x': svar
n': db_index
svar_quantify x' n'
  (if decide (x' = x)
   then patt_bound_svar n'
   else patt_free_svar x) =
(if decide (x' = x)
 then patt_bound_svar n'
 else patt_free_svar x) unfold svar_quantify.
Σ: Signature
x, x': svar
n': db_index
(fix svar_quantify
   (X : svar) (level : db_index) (p : Pattern) {struct
    p} : Pattern :=
   match p with
   | patt_free_evar x' => patt_free_evar x'
   | patt_free_svar X' =>
       if decide (X = X')
       then patt_bound_svar level
       else patt_free_svar X'
   | patt_bound_evar x' => patt_bound_evar x'
   | patt_bound_svar X0 => patt_bound_svar X0
   | patt_sym s => patt_sym s
   | patt_app ls rs =>
       patt_app (svar_quantify X level ls)
         (svar_quantify X level rs)
   | patt_bott => patt_bott
   | patt_imp ls rs =>
       patt_imp (svar_quantify X level ls)
         (svar_quantify X level rs)
   | patt_exists p' =>
       patt_exists (svar_quantify X level p')
   | patt_mu p' =>
       patt_mu (svar_quantify X (S level) p')
   end) x' n'
  (if decide (x' = x)
   then patt_bound_svar n'
   else patt_free_svar x) =
(if decide (x' = x)
 then patt_bound_svar n'
 else patt_free_svar x) repeat case_match; auto.
Σ: Signature
x, x': svar
n': db_index
n: x' ≠ x
e: x' = x
H0: decide (x' = x) = left e
H: left e = right n
patt_bound_svar n' = patt_free_svar x contradiction.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  svar_quantify X n (svar_quantify X n φ1) =
  svar_quantify X n φ1
IHφ2: ∀ (X : svar) (n : db_index),
  svar_quantify X n (svar_quantify X n φ2) =
  svar_quantify X n φ2
x': svar
n': db_index
patt_app
  (svar_quantify x' n' (svar_quantify x' n' φ1))
  (svar_quantify x' n' (svar_quantify x' n' φ2)) =
patt_app (svar_quantify x' n' φ1)
  (svar_quantify x' n' φ2) now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  svar_quantify X n (svar_quantify X n φ1) =
  svar_quantify X n φ1
IHφ2: ∀ (X : svar) (n : db_index),
  svar_quantify X n (svar_quantify X n φ2) =
  svar_quantify X n φ2
x': svar
n': db_index
patt_imp
  (svar_quantify x' n' (svar_quantify x' n' φ1))
  (svar_quantify x' n' (svar_quantify x' n' φ2)) =
patt_imp (svar_quantify x' n' φ1)
  (svar_quantify x' n' φ2) now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (n : db_index),
svar_quantify X n (svar_quantify X n φ) = svar_quantify X n φ
x': svar
n': db_index
patt_exists
  (svar_quantify x' n' (svar_quantify x' n' φ)) =
patt_exists (svar_quantify x' n' φ) now rewrite IHφ.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (n : db_index),
svar_quantify X n (svar_quantify X n φ) = svar_quantify X n φ
x': svar
n': db_index
patt_mu
  (svar_quantify x' (S n') (svar_quantify x' (S n') φ)) =
patt_mu (svar_quantify x' (S n') φ) now rewrite IHφ.
Qed.

Lemma well_formed_bevar_subst φ : forall ψ n m,
m >= n -> well_formed_closed_ex_aux φ n ->
bevar_subst φ ψ m = φ.
Σ: Signature
φ: Pattern
∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
Proof.
Σ: Signature
φ: Pattern
∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
induction φ; intros ψ n' m' H H0; simpl; auto.
Σ: Signature
n: db_index
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_bound_evar n) n'
match compare_nat n m' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) n'
patt_app φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) n'
patt_imp φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_imp φ1 φ2
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ n
    → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_exists φ) n'
patt_exists φ^[evar:S m'↦ψ] = patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ n
    → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_mu φ) n'
patt_mu φ^[evar:m'↦ψ] = patt_mu φ
*
Σ: Signature
n: db_index
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_bound_evar n) n'
match compare_nat n m' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n simpl in H0.
Σ: Signature
n: db_index
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: if decide (n < n') then true else false
match compare_nat n m' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n repeat case_match; auto; try lia; congruence.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) n'
patt_app φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_app φ1 φ2 simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ1 n' &&
well_formed_closed_ex_aux φ2 n'
patt_app φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_app φ1 φ2 apply eq_sym, andb_true_eq in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
∧ true = well_formed_closed_ex_aux φ2 n'
patt_app φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_app φ1 φ2 destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_app φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_app φ1 φ2 erewrite IHφ1, IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_app φ1 φ2 = patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ ?n0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ2 ?n0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ ?n
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ1 ?n
3: apply eq_sym, H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_app φ1 φ2 = patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ ?n
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ1 ?n
4: apply eq_sym, H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_app φ1 φ2 = patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ n' all: auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) n'
patt_imp φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_imp φ1 φ2 simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ1 n' &&
well_formed_closed_ex_aux φ2 n'
patt_imp φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_imp φ1 φ2 apply eq_sym, andb_true_eq in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
∧ true = well_formed_closed_ex_aux φ2 n'
patt_imp φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_imp φ1 φ2 destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_imp φ1^[evar:m'↦ψ] φ2^[evar:m'↦ψ] =
patt_imp φ1 φ2 erewrite IHφ1, IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_imp φ1 φ2 = patt_imp φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ ?n0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ2 ?n0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ ?n
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ1 ?n
3: apply eq_sym, H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_imp φ1 φ2 = patt_imp φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ ?n
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux φ1 ?n
4: apply eq_sym, H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
patt_imp φ1 φ2 = patt_imp φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ1 n
    → φ1^[evar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (n m : nat),
  m ≥ n
  → well_formed_closed_ex_aux φ2 n
    → φ2^[evar:m↦ψ] = φ2
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: true = well_formed_closed_ex_aux φ1 n'
H1: true = well_formed_closed_ex_aux φ2 n'
m' ≥ n' all: auto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_exists φ) n'
patt_exists φ^[evar:S m'↦ψ] = patt_exists φ simpl in H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
patt_exists φ^[evar:S m'↦ψ] = patt_exists φ erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
patt_exists φ = patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
S m' ≥ ?n
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
well_formed_closed_ex_aux φ ?n 3: apply H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
patt_exists φ = patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
S m' ≥ S n' auto.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ (S n')
S m' ≥ S n' lia.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux (patt_mu φ) n'
patt_mu φ^[evar:m'↦ψ] = patt_mu φ simpl in H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ n'
patt_mu φ^[evar:m'↦ψ] = patt_mu φ erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ n'
patt_mu φ = patt_mu φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ n'
m' ≥ ?n
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ n'
well_formed_closed_ex_aux φ ?n 3: apply H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ n'
patt_mu φ = patt_mu φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : nat),
m ≥ n → well_formed_closed_ex_aux φ n → φ^[evar:m↦ψ] = φ
ψ: Pattern
n', m': nat
H: m' ≥ n'
H0: well_formed_closed_ex_aux φ n'
m' ≥ n' all: auto.
Qed.

Lemma well_formed_bsvar_subst φ : forall ψ k m,
m >= k -> well_formed_closed_mu_aux φ k ->
bsvar_subst φ ψ m = φ.
Σ: Signature
φ: Pattern
∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
Proof.
Σ: Signature
φ: Pattern
∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
induction φ; intros ψ k' m' H H0; simpl; auto.
Σ: Signature
n: db_index
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_bound_svar n) k'
match compare_nat n m' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) k'
patt_app φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) k'
patt_imp φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_imp φ1 φ2
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ k
    → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_exists φ) k'
patt_exists φ^[svar:m'↦ψ] = patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ k
    → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_mu φ) k'
patt_mu φ^[svar:S m'↦ψ] = patt_mu φ
*
Σ: Signature
n: db_index
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_bound_svar n) k'
match compare_nat n m' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n simpl in H0.
Σ: Signature
n: db_index
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: if decide (n < k') then true else false
match compare_nat n m' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n repeat case_match; auto; try lia; congruence.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) k'
patt_app φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_app φ1 φ2 simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ1 k' &&
well_formed_closed_mu_aux φ2 k'
patt_app φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_app φ1 φ2 apply eq_sym, andb_true_eq in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
∧ true = well_formed_closed_mu_aux φ2 k'
patt_app φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_app φ1 φ2 destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_app φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_app φ1 φ2 erewrite IHφ1, IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_app φ1 φ2 = patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ ?k0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
well_formed_closed_mu_aux φ2 ?k0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ ?k
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
well_formed_closed_mu_aux φ1 ?k
3: apply eq_sym, H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_app φ1 φ2 = patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ k'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ ?k
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
well_formed_closed_mu_aux φ1 ?k
4: apply eq_sym, H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_app φ1 φ2 = patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ k'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ k' all: auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) k'
patt_imp φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_imp φ1 φ2 simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ1 k' &&
well_formed_closed_mu_aux φ2 k'
patt_imp φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_imp φ1 φ2 apply eq_sym, andb_true_eq in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
∧ true = well_formed_closed_mu_aux φ2 k'
patt_imp φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_imp φ1 φ2 destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_imp φ1^[svar:m'↦ψ] φ2^[svar:m'↦ψ] =
patt_imp φ1 φ2 erewrite IHφ1, IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_imp φ1 φ2 = patt_imp φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ ?k0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
well_formed_closed_mu_aux φ2 ?k0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ ?k
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
well_formed_closed_mu_aux φ1 ?k
3: apply eq_sym, H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_imp φ1 φ2 = patt_imp φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ k'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ ?k
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
well_formed_closed_mu_aux φ1 ?k
4: apply eq_sym, H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
patt_imp φ1 φ2 = patt_imp φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ k'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ1 k
    → φ1^[svar:m↦ψ] = φ1
IHφ2: ∀ (ψ : Pattern) (k m : nat),
  m ≥ k
  → well_formed_closed_mu_aux φ2 k
    → φ2^[svar:m↦ψ] = φ2
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: true = well_formed_closed_mu_aux φ1 k'
H1: true = well_formed_closed_mu_aux φ2 k'
m' ≥ k' all: auto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_exists φ) k'
patt_exists φ^[svar:m'↦ψ] = patt_exists φ simpl in H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
patt_exists φ^[svar:m'↦ψ] = patt_exists φ erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
patt_exists φ = patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
m' ≥ ?k
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
well_formed_closed_mu_aux φ ?k 3: apply H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
patt_exists φ = patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
m' ≥ k' auto.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ k'
m' ≥ k' lia.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux (patt_mu φ) k'
patt_mu φ^[svar:S m'↦ψ] = patt_mu φ simpl in H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
patt_mu φ^[svar:S m'↦ψ] = patt_mu φ erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
patt_mu φ = patt_mu φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
S m' ≥ ?k
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
well_formed_closed_mu_aux φ ?k 3: apply H0.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
patt_mu φ = patt_mu φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
S m' ≥ S k' all: auto.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (k m : nat),
m ≥ k → well_formed_closed_mu_aux φ k → φ^[svar:m↦ψ] = φ
ψ: Pattern
k', m': nat
H: m' ≥ k'
H0: well_formed_closed_mu_aux φ (S k')
S m' ≥ S k' lia.
Qed.

(* bevar_subst is identity if n does not occur in phi *)
Corollary bevar_subst_not_occur n ψ ϕ :
well_formed_closed_ex_aux ϕ n ->
bevar_subst ϕ ψ n = ϕ.
Σ: Signature
n: db_index
ψ, ϕ: Pattern
well_formed_closed_ex_aux ϕ n → ϕ^[evar:n↦ψ] = ϕ
Proof.
Σ: Signature
n: db_index
ψ, ϕ: Pattern
well_formed_closed_ex_aux ϕ n → ϕ^[evar:n↦ψ] = ϕ
intro H.
Σ: Signature
n: db_index
ψ, ϕ: Pattern
H: well_formed_closed_ex_aux ϕ n
ϕ^[evar:n↦ψ] = ϕ eapply well_formed_bevar_subst; eauto.
Qed.

(* evar_open is identity if n does not occur in phi *)
Corollary evar_open_not_occur n x ϕ :
well_formed_closed_ex_aux ϕ n ->
evar_open n x ϕ = ϕ.
Σ: Signature
n: db_index
x: evar
ϕ: Pattern
well_formed_closed_ex_aux ϕ n → evar_open n x ϕ = ϕ
Proof.
Σ: Signature
n: db_index
x: evar
ϕ: Pattern
well_formed_closed_ex_aux ϕ n → evar_open n x ϕ = ϕ
apply bevar_subst_not_occur.
Qed.

(* bsvar_subst is identity if n does not occur in phi *)
Corollary bsvar_subst_not_occur n ψ ϕ :
well_formed_closed_mu_aux ϕ n ->
bsvar_subst ϕ ψ n = ϕ.
Σ: Signature
n: db_index
ψ, ϕ: Pattern
well_formed_closed_mu_aux ϕ n → ϕ^[svar:n↦ψ] = ϕ
Proof.
Σ: Signature
n: db_index
ψ, ϕ: Pattern
well_formed_closed_mu_aux ϕ n → ϕ^[svar:n↦ψ] = ϕ
intro H.
Σ: Signature
n: db_index
ψ, ϕ: Pattern
H: well_formed_closed_mu_aux ϕ n
ϕ^[svar:n↦ψ] = ϕ eapply well_formed_bsvar_subst; eauto.
Qed.

(* evar_open is identity if n does not occur in phi *)
Corollary svar_open_not_occur n x ϕ :
well_formed_closed_mu_aux ϕ n ->
svar_open n x ϕ = ϕ.
Σ: Signature
n: db_index
x: svar
ϕ: Pattern
well_formed_closed_mu_aux ϕ n → svar_open n x ϕ = ϕ
Proof.
Σ: Signature
n: db_index
x: svar
ϕ: Pattern
well_formed_closed_mu_aux ϕ n → svar_open n x ϕ = ϕ
apply bsvar_subst_not_occur.
Qed.

(* opening on closed patterns is identity *)
Lemma evar_open_closed :
forall phi,
well_formed_closed_ex_aux phi 0 ->
forall n v,
  evar_open n v phi = phi.
Σ: Signature
∀ phi : Pattern,
  well_formed_closed_ex_aux phi 0
  → ∀ (n : db_index) (v : evar),
      evar_open n v phi = phi
Proof.
Σ: Signature
∀ phi : Pattern,
  well_formed_closed_ex_aux phi 0
  → ∀ (n : db_index) (v : evar),
      evar_open n v phi = phi
intros phi H n v.
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
evar_open n v phi = phi unfold evar_open.
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
phi^[evar:n↦patt_free_evar v] = phi erewrite well_formed_bevar_subst.
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
phi = phi
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
n ≥ ?n
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
well_formed_closed_ex_aux phi ?n 3: exact H.
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
phi = phi
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
n ≥ 0
auto.
Σ: Signature
phi: Pattern
H: well_formed_closed_ex_aux phi 0
n: db_index
v: evar
n ≥ 0 lia.
Qed.

Lemma svar_open_closed :
forall phi,
well_formed_closed_mu_aux phi 0 ->
forall n v,
  svar_open n v phi = phi.
Σ: Signature
∀ phi : Pattern,
  well_formed_closed_mu_aux phi 0
  → ∀ (n : db_index) (v : svar),
      svar_open n v phi = phi
Proof.
Σ: Signature
∀ phi : Pattern,
  well_formed_closed_mu_aux phi 0
  → ∀ (n : db_index) (v : svar),
      svar_open n v phi = phi 
intros phi H n v.
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
svar_open n v phi = phi unfold svar_open.
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
phi^[svar:n↦patt_free_svar v] = phi erewrite well_formed_bsvar_subst.
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
phi = phi
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
n ≥ ?k
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
well_formed_closed_mu_aux phi ?k 3: exact H.
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
phi = phi
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
n ≥ 0
auto.
Σ: Signature
phi: Pattern
H: well_formed_closed_mu_aux phi 0
n: db_index
v: svar
n ≥ 0 lia.
Qed.

Lemma bevar_subst_comm_higher :
forall phi psi1 psi2 n m, 
n > m -> well_formed_closed_ex_aux psi1 0 -> well_formed_closed_ex_aux psi2 0 ->
bevar_subst (bevar_subst phi psi1 n) psi2 m = 
bevar_subst (bevar_subst phi psi2 m) psi1 (pred n).
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:Init.Nat.pred n↦psi1]
Proof.
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:Init.Nat.pred n↦psi1]
induction phi; intros psi1 psi2 n0 m0 NEQ Hwf1 Hwf2; simpl; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:m0↦psi2] =
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:Init.Nat.pred n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_app phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_app
  phi1^[evar:m0↦psi2]^[evar:
  Init.Nat.pred n0↦psi1]
  phi2^[evar:m0↦psi2]^[evar:
  Init.Nat.pred n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_imp phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_imp
  phi1^[evar:m0↦psi2]^[evar:
  Init.Nat.pred n0↦psi1]
  phi2^[evar:m0↦psi2]^[evar:
  Init.Nat.pred n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_exists phi^[evar:S n0↦psi1]^[evar:S m0↦psi2] =
patt_exists
  phi^[evar:S m0↦psi2]^[evar:
  S (Init.Nat.pred n0)↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_mu phi^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_mu
  phi^[evar:m0↦psi2]^[evar:
  Init.Nat.pred n0↦psi1]
-
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:m0↦psi2] =
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:Init.Nat.pred n0↦psi1] repeat case_match; simpl; try rewrite -> Heqc; try rewrite -> Heqc0; auto; subst; try congruence.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
l0: n < m0
H0: compare_nat n m0 = Nat_less n m0 l0
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: m0 < n0
H: compare_nat m0 n0 = Nat_less m0 n0 l
H0: compare_nat m0 m0 = Nat_equal m0 m0 (erefl m0)
match compare_nat m0 m0 with
| Nat_less _ _ _ => patt_bound_evar m0
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred m0)
end = psi2^[evar:Init.Nat.pred n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
g: n > m0
H0: compare_nat n m0 = Nat_greater n m0 g
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat (Nat.pred n) (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < m0
H0: compare_nat n0 m0 = Nat_less n0 m0 l
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[evar:m0↦psi2] =
match compare_nat n0 (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_evar n0
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n0)
end
Σ: Signature
psi1, psi2: Pattern
m0: nat
NEQ: m0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
H, H0: compare_nat m0 m0 = Nat_equal m0 m0 (erefl m0)
psi1^[evar:m0↦psi2] =
psi2^[evar:Init.Nat.pred m0↦psi1]
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n0 > m0
H0: compare_nat n0 m0 = Nat_greater n0 m0 g
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[evar:m0↦psi2] =
match
  compare_nat (Nat.pred n0) (Init.Nat.pred n0)
with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n0)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n0))
end
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
l: n < m0
H0: compare_nat n m0 = Nat_less n m0 l
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end =
match compare_nat n (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: m0 > n0
H: compare_nat m0 n0 = Nat_greater m0 n0 g
H0: compare_nat m0 m0 = Nat_equal m0 m0 (erefl m0)
match compare_nat (Nat.pred m0) m0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred m0)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred m0))
end = psi2^[evar:Init.Nat.pred n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
g0: n > m0
H0: compare_nat n m0 = Nat_greater n m0 g0
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end =
match compare_nat (Nat.pred n) (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end
all:  repeat case_match; try lia; auto.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: m0 < n0
H: compare_nat m0 n0 = Nat_less m0 n0 l
e: m0 = m0
H1: compare_nat m0 m0 = Nat_equal m0 m0 e
H0: Nat_equal m0 m0 e = Nat_equal m0 m0 (erefl m0)
psi2 = psi2^[evar:Init.Nat.pred n0↦psi1]
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n0 > m0
H0: compare_nat n0 m0 = Nat_greater n0 m0 g
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
e: Nat.pred n0 = Init.Nat.pred n0
H1: compare_nat (Nat.pred n0) (Init.Nat.pred n0) =
Nat_equal (Nat.pred n0) (Init.Nat.pred n0) e
psi1^[evar:m0↦psi2] = psi1
1-2: subst; erewrite well_formed_bevar_subst; try eassumption; auto; lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_app phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_app
  phi1^[evar:m0↦psi2]^[evar:Init.Nat.pred n0↦psi1]
  phi2^[evar:m0↦psi2]^[evar:Init.Nat.pred n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_imp phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_imp
  phi1^[evar:m0↦psi2]^[evar:Init.Nat.pred n0↦psi1]
  phi2^[evar:m0↦psi2]^[evar:Init.Nat.pred n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_exists phi^[evar:S n0↦psi1]^[evar:S m0↦psi2] =
patt_exists
  phi^[evar:S m0↦psi2]^[evar:S (Init.Nat.pred n0)↦psi1] rewrite -> IHphi; auto; try lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_exists
  phi^[evar:S m0↦psi2]^[evar:Init.Nat.pred (S n0)↦psi1] =
patt_exists
  phi^[evar:S m0↦psi2]^[evar:S (Init.Nat.pred n0)↦psi1]
replace (pred (S n0)) with n0 by lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_exists phi^[evar:S m0↦psi2]^[evar:n0↦psi1] =
patt_exists
  phi^[evar:S m0↦psi2]^[evar:S (Init.Nat.pred n0)↦psi1]
now replace (S (pred n0)) with n0 by lia.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:m↦psi2]^[evar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_mu phi^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_mu
  phi^[evar:m0↦psi2]^[evar:Init.Nat.pred n0↦psi1] rewrite -> IHphi; auto.
Qed.

Lemma bevar_subst_comm_lower :
forall phi psi1 psi2 n m, 
n < m -> well_formed_closed_ex_aux psi1 0 -> well_formed_closed_ex_aux psi2 0 ->
bevar_subst (bevar_subst phi psi1 n) psi2 m = 
bevar_subst (bevar_subst phi psi2 (S m)) psi1 n.
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:S m↦psi2]^[evar:n↦psi1]
Proof.
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:S m↦psi2]^[evar:n↦psi1]
induction phi; intros psi1 psi2 n0 m0 NEQ Hwf1 Hwf2; simpl; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:m0↦psi2] =
match compare_nat n (S m0) with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:S m↦psi2]^[evar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_app phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_app phi1^[evar:S m0↦psi2]^[evar:n0↦psi1]
  phi2^[evar:S m0↦psi2]^[evar:n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:S m↦psi2]^[evar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_imp phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_imp phi1^[evar:S m0↦psi2]^[evar:n0↦psi1]
  phi2^[evar:S m0↦psi2]^[evar:n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_exists phi^[evar:S n0↦psi1]^[evar:S m0↦psi2] =
patt_exists phi^[evar:S (S m0)↦psi2]^[evar:S n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_mu phi^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_mu phi^[evar:S m0↦psi2]^[evar:n0↦psi1]
-
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:m0↦psi2] =
match compare_nat n (S m0) with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:n0↦psi1] repeat case_match; simpl; try rewrite -> Heqc; try rewrite -> Heqc0; auto; subst; try congruence.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
l0: n < S m0
H0: compare_nat n (S m0) = Nat_less n (S m0) l0
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: S m0 < n0
H: compare_nat (S m0) n0 = Nat_less (S m0) n0 l
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
match compare_nat (S m0) m0 with
| Nat_less _ _ _ => patt_bound_evar (S m0)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (S m0))
end = psi2^[evar:n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
g: n > S m0
H0: compare_nat n (S m0) = Nat_greater n (S m0) g
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat (Nat.pred n) n0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[evar:m0↦psi2] =
match compare_nat n0 n0 with
| Nat_less _ _ _ => patt_bound_evar n0
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n0)
end
Σ: Signature
psi1, psi2: Pattern
m0: nat
NEQ: S m0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
H, H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
psi1^[evar:m0↦psi2] = psi2^[evar:S m0↦psi1]
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n0 > S m0
H0: compare_nat n0 (S m0) = Nat_greater n0 (S m0) g
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[evar:m0↦psi2] =
match compare_nat (Nat.pred n0) n0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n0)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n0))
end
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
l: n < S m0
H0: compare_nat n (S m0) = Nat_less n (S m0) l
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end =
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: S m0 > n0
H: compare_nat (S m0) n0 = Nat_greater (S m0) n0 g
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
match compare_nat (Nat.pred (S m0)) m0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred (S m0))
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred (S m0)))
end = psi2^[evar:n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
g0: n > S m0
H0: compare_nat n (S m0) = Nat_greater n (S m0) g0
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end =
match compare_nat (Nat.pred n) n0 with
| Nat_less _ _ _ => patt_bound_evar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_evar (Nat.pred (Nat.pred n))
end
all:  repeat case_match; try lia; auto.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
psi1^[evar:m0↦psi2] = psi1
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: S m0 > n0
H: compare_nat (S m0) n0 = Nat_greater (S m0) n0 g
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
e: Nat.pred (S m0) = m0
H1: compare_nat (Nat.pred (S m0)) m0 =
Nat_equal (Nat.pred (S m0)) m0 e
psi2 = psi2^[evar:n0↦psi1]
1-2: subst; erewrite well_formed_bevar_subst; try eassumption; auto.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
well_formed_closed_ex_aux psi1 (S n0)
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
g: S m0 > n0
H: compare_nat (S m0) n0 = Nat_greater (S m0) n0 g
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
e: Nat.pred (S m0) = m0
H1: compare_nat (Nat.pred (S m0)) m0 =
Nat_equal (Nat.pred (S m0)) m0 e
n0 ≥ 0 2: lia.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
well_formed_closed_ex_aux psi1 (S n0)
eapply well_formed_closed_ex_aux_ind.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
?ind_evar1 ≤ S n0
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
well_formed_closed_ex_aux psi1 ?ind_evar1 = true 2: exact Hwf1.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
0 ≤ S n0 lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:S m↦psi2]^[evar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_app phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_app phi1^[evar:S m0↦psi2]^[evar:n0↦psi1]
  phi2^[evar:S m0↦psi2]^[evar:n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi1^[evar:n↦psi1]^[evar:m↦psi2] =
        phi1^[evar:S m↦psi2]^[evar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi2^[evar:n↦psi1]^[evar:m↦psi2] =
        phi2^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_imp phi1^[evar:n0↦psi1]^[evar:m0↦psi2]
  phi2^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_imp phi1^[evar:S m0↦psi2]^[evar:n0↦psi1]
  phi2^[evar:S m0↦psi2]^[evar:n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_exists phi^[evar:S n0↦psi1]^[evar:S m0↦psi2] =
patt_exists phi^[evar:S (S m0)↦psi2]^[evar:S n0↦psi1] rewrite -> IHphi; auto; try lia.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_ex_aux psi1 0
    → well_formed_closed_ex_aux psi2 0
      → phi^[evar:n↦psi1]^[evar:m↦psi2] =
        phi^[evar:S m↦psi2]^[evar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_ex_aux psi2 0
patt_mu phi^[evar:n0↦psi1]^[evar:m0↦psi2] =
patt_mu phi^[evar:S m0↦psi2]^[evar:n0↦psi1] rewrite -> IHphi; auto.
Qed.

Lemma bsvar_subst_comm_higher :
forall phi psi1 psi2 n m, 
n > m -> well_formed_closed_mu_aux psi1 0 -> well_formed_closed_mu_aux psi2 0 ->
bsvar_subst (bsvar_subst phi psi1 n) psi2 m = 
bsvar_subst (bsvar_subst phi psi2 m) psi1 (pred n).
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:Init.Nat.pred n↦psi1]
Proof.
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:Init.Nat.pred n↦psi1]
induction phi; intros psi1 psi2 n0 m0 NEQ Hwf1 Hwf2; simpl; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:m0↦psi2] =
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:Init.Nat.pred n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_app phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_app
  phi1^[svar:m0↦psi2]^[svar:
  Init.Nat.pred n0↦psi1]
  phi2^[svar:m0↦psi2]^[svar:
  Init.Nat.pred n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_imp phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_imp
  phi1^[svar:m0↦psi2]^[svar:
  Init.Nat.pred n0↦psi1]
  phi2^[svar:m0↦psi2]^[svar:
  Init.Nat.pred n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_exists phi^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_exists
  phi^[svar:m0↦psi2]^[svar:
  Init.Nat.pred n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu phi^[svar:S n0↦psi1]^[svar:S m0↦psi2] =
patt_mu
  phi^[svar:S m0↦psi2]^[svar:
  S (Init.Nat.pred n0)↦psi1]
-
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:m0↦psi2] =
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:Init.Nat.pred n0↦psi1] repeat case_match; simpl; try rewrite -> Heqc; try rewrite -> Heqc0; auto; subst; try congruence.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
l0: n < m0
H0: compare_nat n m0 = Nat_less n m0 l0
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end =
match compare_nat n (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: m0 < n0
H: compare_nat m0 n0 = Nat_less m0 n0 l
H0: compare_nat m0 m0 = Nat_equal m0 m0 (erefl m0)
match compare_nat m0 m0 with
| Nat_less _ _ _ => patt_bound_svar m0
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred m0)
end = psi2^[svar:Init.Nat.pred n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
g: n > m0
H0: compare_nat n m0 = Nat_greater n m0 g
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end =
match compare_nat (Nat.pred n) (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < m0
H0: compare_nat n0 m0 = Nat_less n0 m0 l
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[svar:m0↦psi2] =
match compare_nat n0 (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_svar n0
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n0)
end
Σ: Signature
psi1, psi2: Pattern
m0: nat
NEQ: m0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
H, H0: compare_nat m0 m0 = Nat_equal m0 m0 (erefl m0)
psi1^[svar:m0↦psi2] =
psi2^[svar:Init.Nat.pred m0↦psi1]
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n0 > m0
H0: compare_nat n0 m0 = Nat_greater n0 m0 g
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[svar:m0↦psi2] =
match
  compare_nat (Nat.pred n0) (Init.Nat.pred n0)
with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n0)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n0))
end
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
l: n < m0
H0: compare_nat n m0 = Nat_less n m0 l
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end =
match compare_nat n (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: m0 > n0
H: compare_nat m0 n0 = Nat_greater m0 n0 g
H0: compare_nat m0 m0 = Nat_equal m0 m0 (erefl m0)
match compare_nat (Nat.pred m0) m0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred m0)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred m0))
end = psi2^[svar:Init.Nat.pred n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
g0: n > m0
H0: compare_nat n m0 = Nat_greater n m0 g0
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end =
match compare_nat (Nat.pred n) (Init.Nat.pred n0) with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end
all:  repeat case_match; try lia; auto.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: m0 < n0
H: compare_nat m0 n0 = Nat_less m0 n0 l
e: m0 = m0
H1: compare_nat m0 m0 = Nat_equal m0 m0 e
H0: Nat_equal m0 m0 e = Nat_equal m0 m0 (erefl m0)
psi2 = psi2^[svar:Init.Nat.pred n0↦psi1]
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n0 > m0
H0: compare_nat n0 m0 = Nat_greater n0 m0 g
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
e: Nat.pred n0 = Init.Nat.pred n0
H1: compare_nat (Nat.pred n0) (Init.Nat.pred n0) =
Nat_equal (Nat.pred n0) (Init.Nat.pred n0) e
psi1^[svar:m0↦psi2] = psi1
1-2: subst; erewrite well_formed_bsvar_subst; try eassumption; auto; lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_app phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_app
  phi1^[svar:m0↦psi2]^[svar:Init.Nat.pred n0↦psi1]
  phi2^[svar:m0↦psi2]^[svar:Init.Nat.pred n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_imp phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_imp
  phi1^[svar:m0↦psi2]^[svar:Init.Nat.pred n0↦psi1]
  phi2^[svar:m0↦psi2]^[svar:Init.Nat.pred n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_exists phi^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_exists
  phi^[svar:m0↦psi2]^[svar:Init.Nat.pred n0↦psi1] rewrite -> IHphi; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu phi^[svar:S n0↦psi1]^[svar:S m0↦psi2] =
patt_mu
  phi^[svar:S m0↦psi2]^[svar:S (Init.Nat.pred n0)↦psi1] rewrite -> IHphi; auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu
  phi^[svar:S m0↦psi2]^[svar:Init.Nat.pred (S n0)↦psi1] =
patt_mu
  phi^[svar:S m0↦psi2]^[svar:S (Init.Nat.pred n0)↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
S n0 > S m0 2: lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu
  phi^[svar:S m0↦psi2]^[svar:Init.Nat.pred (S n0)↦psi1] =
patt_mu
  phi^[svar:S m0↦psi2]^[svar:S (Init.Nat.pred n0)↦psi1]
replace (pred (S n0)) with n0 by lia.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n > m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:m↦psi2]^[svar:
        Init.Nat.pred n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 > m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu phi^[svar:S m0↦psi2]^[svar:n0↦psi1] =
patt_mu
  phi^[svar:S m0↦psi2]^[svar:S (Init.Nat.pred n0)↦psi1]
now replace (S (pred n0)) with n0 by lia.
Qed.

Lemma bsvar_subst_comm_lower :
forall phi psi1 psi2 n m, 
n < m -> well_formed_closed_mu_aux psi1 0 -> well_formed_closed_mu_aux psi2 0 ->
bsvar_subst (bsvar_subst phi psi1 n) psi2 m = 
bsvar_subst (bsvar_subst phi psi2 (S m)) psi1 n.
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
Proof.
Σ: Signature
∀ (phi psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
induction phi; intros psi1 psi2 n0 m0 NEQ Hwf1 Hwf2; simpl; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:m0↦psi2] =
match compare_nat n (S m0) with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:S m↦psi2]^[svar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_app phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_app phi1^[svar:S m0↦psi2]^[svar:n0↦psi1]
  phi2^[svar:S m0↦psi2]^[svar:n0↦psi1]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:S m↦psi2]^[svar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_imp phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_imp phi1^[svar:S m0↦psi2]^[svar:n0↦psi1]
  phi2^[svar:S m0↦psi2]^[svar:n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_exists phi^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_exists phi^[svar:S m0↦psi2]^[svar:n0↦psi1]
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu phi^[svar:S n0↦psi1]^[svar:S m0↦psi2] =
patt_mu phi^[svar:S (S m0)↦psi2]^[svar:S n0↦psi1]
-
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:m0↦psi2] =
match compare_nat n (S m0) with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[svar:n0↦psi1] repeat case_match; simpl; try rewrite -> Heqc; try rewrite -> Heqc0; auto; subst; try congruence.
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
l0: n < S m0
H0: compare_nat n (S m0) = Nat_less n (S m0) l0
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end =
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: S m0 < n0
H: compare_nat (S m0) n0 = Nat_less (S m0) n0 l
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
match compare_nat (S m0) m0 with
| Nat_less _ _ _ => patt_bound_svar (S m0)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (S m0))
end = psi2^[svar:n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n < n0
H: compare_nat n n0 = Nat_less n n0 l
g: n > S m0
H0: compare_nat n (S m0) = Nat_greater n (S m0) g
match compare_nat n m0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end =
match compare_nat (Nat.pred n) n0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[svar:m0↦psi2] =
match compare_nat n0 n0 with
| Nat_less _ _ _ => patt_bound_svar n0
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n0)
end
Σ: Signature
psi1, psi2: Pattern
m0: nat
NEQ: S m0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
H, H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
psi1^[svar:m0↦psi2] = psi2^[svar:S m0↦psi1]
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n0 > S m0
H0: compare_nat n0 (S m0) = Nat_greater n0 (S m0) g
H: compare_nat n0 n0 = Nat_equal n0 n0 (erefl n0)
psi1^[svar:m0↦psi2] =
match compare_nat (Nat.pred n0) n0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n0)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n0))
end
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
l: n < S m0
H0: compare_nat n (S m0) = Nat_less n (S m0) l
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end =
match compare_nat n n0 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: S m0 > n0
H: compare_nat (S m0) n0 = Nat_greater (S m0) n0 g
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
match compare_nat (Nat.pred (S m0)) m0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred (S m0))
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred (S m0)))
end = psi2^[svar:n0↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: n > n0
H: compare_nat n n0 = Nat_greater n n0 g
g0: n > S m0
H0: compare_nat n (S m0) = Nat_greater n (S m0) g0
match compare_nat (Nat.pred n) m0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end =
match compare_nat (Nat.pred n) n0 with
| Nat_less _ _ _ => patt_bound_svar (Nat.pred n)
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ =>
    patt_bound_svar (Nat.pred (Nat.pred n))
end
all:  repeat case_match; try lia; auto.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
psi1^[svar:m0↦psi2] = psi1
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: S m0 > n0
H: compare_nat (S m0) n0 = Nat_greater (S m0) n0 g
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
e: Nat.pred (S m0) = m0
H1: compare_nat (Nat.pred (S m0)) m0 =
Nat_equal (Nat.pred (S m0)) m0 e
psi2 = psi2^[svar:n0↦psi1]
1-2: subst; erewrite well_formed_bsvar_subst; try eassumption; auto.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
well_formed_closed_mu_aux psi1 (S n0)
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
g: S m0 > n0
H: compare_nat (S m0) n0 = Nat_greater (S m0) n0 g
H0: compare_nat (S m0) (S m0) =
Nat_equal (S m0) (S m0) (erefl (S m0))
e: Nat.pred (S m0) = m0
H1: compare_nat (Nat.pred (S m0)) m0 =
Nat_equal (Nat.pred (S m0)) m0 e
n0 ≥ 0 2: lia.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
well_formed_closed_mu_aux psi1 (S n0)
eapply well_formed_closed_mu_aux_ind.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
?ind_svar1 ≤ S n0
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
well_formed_closed_mu_aux psi1 ?ind_svar1 = true 2: exact Hwf1.
Σ: Signature
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
l: n0 < S m0
H0: compare_nat n0 (S m0) = Nat_less n0 (S m0) l
e: n0 = n0
H1: compare_nat n0 n0 = Nat_equal n0 n0 e
H: Nat_equal n0 n0 e = Nat_equal n0 n0 (erefl n0)
0 ≤ S n0 lia.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:S m↦psi2]^[svar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_app phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_app phi1^[svar:S m0↦psi2]^[svar:n0↦psi1]
  phi2^[svar:S m0↦psi2]^[svar:n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi1^[svar:n↦psi1]^[svar:m↦psi2] =
        phi1^[svar:S m↦psi2]^[svar:n↦psi1]
IHphi2: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi2^[svar:n↦psi1]^[svar:m↦psi2] =
        phi2^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_imp phi1^[svar:n0↦psi1]^[svar:m0↦psi2]
  phi2^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_imp phi1^[svar:S m0↦psi2]^[svar:n0↦psi1]
  phi2^[svar:S m0↦psi2]^[svar:n0↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_exists phi^[svar:n0↦psi1]^[svar:m0↦psi2] =
patt_exists phi^[svar:S m0↦psi2]^[svar:n0↦psi1] rewrite -> IHphi; auto.
-
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
patt_mu phi^[svar:S n0↦psi1]^[svar:S m0↦psi2] =
patt_mu phi^[svar:S (S m0)↦psi2]^[svar:S n0↦psi1] rewrite -> IHphi; auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (psi1 psi2 : Pattern) (n m : nat),
  n < m
  → well_formed_closed_mu_aux psi1 0
    → well_formed_closed_mu_aux psi2 0
      → phi^[svar:n↦psi1]^[svar:m↦psi2] =
        phi^[svar:S m↦psi2]^[svar:n↦psi1]
psi1, psi2: Pattern
n0, m0: nat
NEQ: n0 < m0
Hwf1: well_formed_closed_mu_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0
S n0 < S m0 lia.
Qed.

Corollary evar_open_comm_higher:
forall n m,
n < m 
->
forall x y phi,
  evar_open n x (evar_open m y phi) = evar_open (pred m) y (evar_open n x phi).
Σ: Signature
∀ n m : nat,
  n < m
  → ∀ (x y : evar) (phi : Pattern),
      evar_open n x (evar_open m y phi) =
      evar_open (Init.Nat.pred m) y
        (evar_open n x phi)
Proof.
Σ: Signature
∀ n m : nat,
  n < m
  → ∀ (x y : evar) (phi : Pattern),
      evar_open n x (evar_open m y phi) =
      evar_open (Init.Nat.pred m) y
        (evar_open n x phi)
intros n m Hneqnm x y phi.
Σ: Signature
n, m: nat
Hneqnm: n < m
x, y: evar
phi: Pattern
evar_open n x (evar_open m y phi) =
evar_open (Init.Nat.pred m) y (evar_open n x phi) apply bevar_subst_comm_higher; auto.
Qed.

Corollary evar_open_comm_lower:
forall n m,
n > m 
->
forall x y phi,
  evar_open n x (evar_open m y phi) = evar_open m y (evar_open (S n) x phi).
Σ: Signature
∀ n m : nat,
  n > m
  → ∀ (x y : evar) (phi : Pattern),
      evar_open n x (evar_open m y phi) =
      evar_open m y (evar_open (S n) x phi)
Proof.
Σ: Signature
∀ n m : nat,
  n > m
  → ∀ (x y : evar) (phi : Pattern),
      evar_open n x (evar_open m y phi) =
      evar_open m y (evar_open (S n) x phi)
intros n m Hneqnm x y phi.
Σ: Signature
n, m: nat
Hneqnm: n > m
x, y: evar
phi: Pattern
evar_open n x (evar_open m y phi) =
evar_open m y (evar_open (S n) x phi) apply bevar_subst_comm_lower; auto.
Qed.

Corollary svar_open_comm_higher:
forall n m,
n < m 
->
forall X Y phi,
  svar_open n X (svar_open m Y phi) = svar_open (pred m) Y (svar_open n X phi).
Σ: Signature
∀ n m : nat,
  n < m
  → ∀ (X Y : svar) (phi : Pattern),
      svar_open n X (svar_open m Y phi) =
      svar_open (Init.Nat.pred m) Y
        (svar_open n X phi)
Proof.
Σ: Signature
∀ n m : nat,
  n < m
  → ∀ (X Y : svar) (phi : Pattern),
      svar_open n X (svar_open m Y phi) =
      svar_open (Init.Nat.pred m) Y
        (svar_open n X phi)
intros n m Hneqnm x y phi.
Σ: Signature
n, m: nat
Hneqnm: n < m
x, y: svar
phi: Pattern
svar_open n x (svar_open m y phi) =
svar_open (Init.Nat.pred m) y (svar_open n x phi) apply bsvar_subst_comm_higher; auto.
Qed.

Corollary svar_open_comm_lower:
forall n m,
n > m
->
forall X Y phi,
  svar_open n X (svar_open m Y phi) = svar_open m Y (svar_open (S n) X phi).
Σ: Signature
∀ n m : nat,
  n > m
  → ∀ (X Y : svar) (phi : Pattern),
      svar_open n X (svar_open m Y phi) =
      svar_open m Y (svar_open (S n) X phi)
Proof.
Σ: Signature
∀ n m : nat,
  n > m
  → ∀ (X Y : svar) (phi : Pattern),
      svar_open n X (svar_open m Y phi) =
      svar_open m Y (svar_open (S n) X phi)
intros n m Hneqnm x y phi.
Σ: Signature
n, m: nat
Hneqnm: n > m
x, y: svar
phi: Pattern
svar_open n x (svar_open m y phi) =
svar_open m y (svar_open (S n) x phi) apply bsvar_subst_comm_lower; auto.
Qed.

Lemma bevar_subst_bsvar_subst phi psi1 psi2 dbi1 dbi2
: well_formed_closed psi1 -> well_formed_closed psi2 ->
bevar_subst (bsvar_subst phi psi1 dbi1) psi2 dbi2
= bsvar_subst (bevar_subst phi psi2 dbi2) psi1 dbi1.
Σ: Signature
phi, psi1, psi2: Pattern
dbi1, dbi2: db_index
well_formed_closed psi1
→ well_formed_closed psi2
  → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
    phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
Proof.
Σ: Signature
phi, psi1, psi2: Pattern
dbi1, dbi2: db_index
well_formed_closed psi1
→ well_formed_closed psi2
  → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
    phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
generalize dependent dbi1.
Σ: Signature
phi, psi1, psi2: Pattern
dbi2: db_index
∀ dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1] generalize dependent dbi2.
Σ: Signature
phi, psi1, psi2: Pattern
∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
induction phi; intros dbi1 dbi2 Hwf1 Hwf2; simpl; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
match compare_nat n dbi1 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n dbi1 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[svar:dbi2↦psi1]
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
match compare_nat n dbi2 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[evar:dbi1↦psi2] =
match compare_nat n dbi2 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end
Σ: Signature
phi1, phi2, psi1, psi2: Pattern
IHphi1: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi1^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi1^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
IHphi2: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi2^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi2^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_app phi1^[svar:dbi2↦psi1]^[evar:dbi1↦psi2]
  phi2^[svar:dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_app phi1^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
  phi2^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
Σ: Signature
phi1, phi2, psi1, psi2: Pattern
IHphi1: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi1^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi1^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
IHphi2: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi2^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi2^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_imp phi1^[svar:dbi2↦psi1]^[evar:dbi1↦psi2]
  phi2^[svar:dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_imp phi1^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
  phi2^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
Σ: Signature
phi, psi1, psi2: Pattern
IHphi: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_exists phi^[svar:dbi2↦psi1]^[evar:S dbi1↦psi2] =
patt_exists phi^[evar:S dbi1↦psi2]^[svar:dbi2↦psi1]
Σ: Signature
phi, psi1, psi2: Pattern
IHphi: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_mu phi^[svar:S dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_mu phi^[evar:dbi1↦psi2]^[svar:S dbi2↦psi1]
*
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
match compare_nat n dbi1 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n dbi1 with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => psi2
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[svar:dbi2↦psi1] break_match_goal; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
psi2 = psi2^[svar:dbi2↦psi1] erewrite well_formed_bsvar_subst; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
well_formed_closed_mu_aux psi2 dbi2
unfold well_formed_closed in *.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed_mu_aux psi1 0 &&
well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0 &&
well_formed_closed_ex_aux psi2 0
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
well_formed_closed_mu_aux psi2 dbi2 destruct_and!.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
well_formed_closed_mu_aux psi2 dbi2
eapply well_formed_closed_mu_aux_ind.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
?ind_svar1 ≤ dbi2
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
well_formed_closed_mu_aux psi2 ?ind_svar1 = true 2: eassumption.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi1
Heqc: compare_nat n dbi1 = Nat_equal n dbi1 e
0 ≤ dbi2 lia.
*
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
match compare_nat n dbi2 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[evar:dbi1↦psi2] =
match compare_nat n dbi2 with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => psi1
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end break_match_goal; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
psi1^[evar:dbi1↦psi2] = psi1 erewrite well_formed_bevar_subst; auto.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
well_formed_closed_ex_aux psi1 dbi1
unfold well_formed_closed in *.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
Hwf1: well_formed_closed_mu_aux psi1 0 &&
well_formed_closed_ex_aux psi1 0
Hwf2: well_formed_closed_mu_aux psi2 0 &&
well_formed_closed_ex_aux psi2 0
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
well_formed_closed_ex_aux psi1 dbi1 destruct_and!.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
well_formed_closed_ex_aux psi1 dbi1
eapply well_formed_closed_ex_aux_ind.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
?ind_evar1 ≤ dbi1
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
well_formed_closed_ex_aux psi1 ?ind_evar1 = true 2: eassumption.
Σ: Signature
n: db_index
psi1, psi2: Pattern
dbi1, dbi2: db_index
H1: well_formed_closed_mu_aux psi1 0 = true
H2: well_formed_closed_ex_aux psi1 0 = true
H: well_formed_closed_mu_aux psi2 0 = true
H0: well_formed_closed_ex_aux psi2 0 = true
e: n = dbi2
Heqc: compare_nat n dbi2 = Nat_equal n dbi2 e
0 ≤ dbi1 lia.
*
Σ: Signature
phi1, phi2, psi1, psi2: Pattern
IHphi1: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi1^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi1^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
IHphi2: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi2^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi2^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_app phi1^[svar:dbi2↦psi1]^[evar:dbi1↦psi2]
  phi2^[svar:dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_app phi1^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
  phi2^[evar:dbi1↦psi2]^[svar:dbi2↦psi1] simpl.
Σ: Signature
phi1, phi2, psi1, psi2: Pattern
IHphi1: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi1^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi1^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
IHphi2: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi2^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi2^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_app phi1^[svar:dbi2↦psi1]^[evar:dbi1↦psi2]
  phi2^[svar:dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_app phi1^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
  phi2^[evar:dbi1↦psi2]^[svar:dbi2↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
*
Σ: Signature
phi1, phi2, psi1, psi2: Pattern
IHphi1: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi1^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi1^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
IHphi2: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi2^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi2^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_imp phi1^[svar:dbi2↦psi1]^[evar:dbi1↦psi2]
  phi2^[svar:dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_imp phi1^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
  phi2^[evar:dbi1↦psi2]^[svar:dbi2↦psi1] simpl.
Σ: Signature
phi1, phi2, psi1, psi2: Pattern
IHphi1: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi1^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi1^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
IHphi2: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi2^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi2^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_imp phi1^[svar:dbi2↦psi1]^[evar:dbi1↦psi2]
  phi2^[svar:dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_imp phi1^[evar:dbi1↦psi2]^[svar:dbi2↦psi1]
  phi2^[evar:dbi1↦psi2]^[svar:dbi2↦psi1] rewrite -> IHphi1, -> IHphi2; auto.
*
Σ: Signature
phi, psi1, psi2: Pattern
IHphi: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_exists phi^[svar:dbi2↦psi1]^[evar:S dbi1↦psi2] =
patt_exists phi^[evar:S dbi1↦psi2]^[svar:dbi2↦psi1] simpl.
Σ: Signature
phi, psi1, psi2: Pattern
IHphi: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_exists phi^[svar:dbi2↦psi1]^[evar:S dbi1↦psi2] =
patt_exists phi^[evar:S dbi1↦psi2]^[svar:dbi2↦psi1] rewrite IHphi; auto.
*
Σ: Signature
phi, psi1, psi2: Pattern
IHphi: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_mu phi^[svar:S dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_mu phi^[evar:dbi1↦psi2]^[svar:S dbi2↦psi1] simpl.
Σ: Signature
phi, psi1, psi2: Pattern
IHphi: ∀ dbi2 dbi1 : db_index,
  well_formed_closed psi1
  → well_formed_closed psi2
    → phi^[svar:dbi1↦psi1]^[evar:dbi2↦psi2] =
      phi^[evar:dbi2↦psi2]^[svar:dbi1↦psi1]
dbi1, dbi2: db_index
Hwf1: well_formed_closed psi1
Hwf2: well_formed_closed psi2
patt_mu phi^[svar:S dbi2↦psi1]^[evar:dbi1↦psi2] =
patt_mu phi^[evar:dbi1↦psi2]^[svar:S dbi2↦psi1] rewrite IHphi; auto.
Qed.

Corollary svar_open_evar_open_comm
: forall (phi : Pattern) (dbi1 : db_index)(x : evar)(dbi2 : db_index)(X : svar),
evar_open dbi1 x (svar_open dbi2 X phi) = svar_open dbi2 X (evar_open dbi1 x phi).
Σ: Signature
∀ (phi : Pattern) (dbi1 : db_index) (x : evar) (dbi2 : db_index) 
  (X : svar),
  evar_open dbi1 x (svar_open dbi2 X phi) =
  svar_open dbi2 X (evar_open dbi1 x phi)
Proof.
Σ: Signature
∀ (phi : Pattern) (dbi1 : db_index) (x : evar) (dbi2 : db_index) 
  (X : svar),
  evar_open dbi1 x (svar_open dbi2 X phi) =
  svar_open dbi2 X (evar_open dbi1 x phi)
intros phi dbi1 x dbi2 X.
Σ: Signature
phi: Pattern
dbi1: db_index
x: evar
dbi2: db_index
X: svar
evar_open dbi1 x (svar_open dbi2 X phi) =
svar_open dbi2 X (evar_open dbi1 x phi) apply bevar_subst_bsvar_subst; auto.
Qed.


Lemma free_svars_evar_open : forall (ϕ : Pattern) (dbi :db_index) (x : evar),
free_svars (evar_open dbi x ϕ) = free_svars ϕ.
Σ: Signature
∀ (ϕ : Pattern) (dbi : db_index) (x : evar),
  free_svars (evar_open dbi x ϕ) = free_svars ϕ
Proof.
Σ: Signature
∀ (ϕ : Pattern) (dbi : db_index) (x : evar),
  free_svars (evar_open dbi x ϕ) = free_svars ϕ
unfold evar_open.
Σ: Signature
∀ (ϕ : Pattern) (dbi : db_index) (x : evar),
  free_svars ϕ^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ
induction ϕ; intros dbi x'; simpl; try reflexivity.
Σ: Signature
n, dbi: db_index
x': evar
free_svars
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x'
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = ∅
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1^[evar:dbi↦patt_free_evar x']
∪ free_svars ϕ2^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ1 ∪ free_svars ϕ2
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1^[evar:dbi↦patt_free_evar x']
∪ free_svars ϕ2^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ1 ∪ free_svars ϕ2
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ
dbi: db_index
x': evar
free_svars ϕ^[evar:S dbi↦patt_free_evar x'] =
free_svars ϕ
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ
dbi: db_index
x': evar
free_svars ϕ^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ
*
Σ: Signature
n, dbi: db_index
x': evar
free_svars
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x'
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = ∅ case_match; reflexivity.
*
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1^[evar:dbi↦patt_free_evar x']
∪ free_svars ϕ2^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ1 ∪ free_svars ϕ2 rewrite -> IHϕ1.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1
∪ free_svars ϕ2^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ1 ∪ free_svars ϕ2 rewrite -> IHϕ2.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1 ∪ free_svars ϕ2 =
free_svars ϕ1 ∪ free_svars ϕ2 reflexivity.
*
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1^[evar:dbi↦patt_free_evar x']
∪ free_svars ϕ2^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ1 ∪ free_svars ϕ2 rewrite -> IHϕ1.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1
∪ free_svars ϕ2^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ1 ∪ free_svars ϕ2 rewrite -> IHϕ2.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ1^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ1
IHϕ2: ∀ (dbi : db_index) (x : evar),
  free_svars ϕ2^[evar:dbi↦patt_free_evar x] =
  free_svars ϕ2
dbi: db_index
x': evar
free_svars ϕ1 ∪ free_svars ϕ2 =
free_svars ϕ1 ∪ free_svars ϕ2 reflexivity.
*
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (dbi : db_index) (x : evar),
free_svars ϕ^[evar:dbi↦patt_free_evar x] = free_svars ϕ
dbi: db_index
x': evar
free_svars ϕ^[evar:S dbi↦patt_free_evar x'] =
free_svars ϕ rewrite -> IHϕ.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (dbi : db_index) (x : evar),
free_svars ϕ^[evar:dbi↦patt_free_evar x] = free_svars ϕ
dbi: db_index
x': evar
free_svars ϕ = free_svars ϕ reflexivity.
*
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (dbi : db_index) (x : evar),
free_svars ϕ^[evar:dbi↦patt_free_evar x] = free_svars ϕ
dbi: db_index
x': evar
free_svars ϕ^[evar:dbi↦patt_free_evar x'] =
free_svars ϕ rewrite -> IHϕ.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (dbi : db_index) (x : evar),
free_svars ϕ^[evar:dbi↦patt_free_evar x] = free_svars ϕ
dbi: db_index
x': evar
free_svars ϕ = free_svars ϕ reflexivity.
Qed.


Lemma positive_negative_occurrence_db_named :
forall (phi : Pattern) (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi ->
   svar_has_positive_occurrence X phi = false ->
   svar_has_positive_occurrence X (svar_open dbi X phi) = false)
  /\ (no_negative_occurrence_db_b dbi phi ->
      svar_has_negative_occurrence X phi = false ->
      svar_has_negative_occurrence X (svar_open dbi X phi) = false).
Σ: Signature
∀ (phi : Pattern) (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi
   → svar_has_positive_occurrence X phi = false
     → svar_has_positive_occurrence X
         (svar_open dbi X phi) = false)
  ∧ (no_negative_occurrence_db_b dbi phi
     → svar_has_negative_occurrence X phi = false
       → svar_has_negative_occurrence X
           (svar_open dbi X phi) = false)
Proof.
Σ: Signature
∀ (phi : Pattern) (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi
   → svar_has_positive_occurrence X phi = false
     → svar_has_positive_occurrence X
         (svar_open dbi X phi) = false)
  ∧ (no_negative_occurrence_db_b dbi phi
     → svar_has_negative_occurrence X phi = false
       → svar_has_negative_occurrence X
           (svar_open dbi X phi) = false)
unfold svar_open.
Σ: Signature
∀ (phi : Pattern) (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi
   → svar_has_positive_occurrence X phi = false
     → svar_has_positive_occurrence X
         phi^[svar:dbi↦patt_free_svar X] = false)
  ∧ (no_negative_occurrence_db_b dbi phi
     → svar_has_negative_occurrence X phi = false
       → svar_has_negative_occurrence X
           phi^[svar:dbi↦patt_free_svar X] = false)
induction phi; intros dbi X; split; simpl; try firstorder; cbn in *.
Σ: Signature
n, dbi: db_index
X: svar
H: if decide (n = dbi) then false else true
H0: false = false
svar_has_positive_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false
Σ: Signature
n, dbi: db_index
X: svar
H: true
H0: false = false
svar_has_negative_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: no_positive_occurrence_db_b dbi phi1 &&
no_positive_occurrence_db_b dbi phi2
H0: svar_has_positive_occurrence X phi1
|| svar_has_positive_occurrence X phi2 = false
svar_has_positive_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: no_negative_occurrence_db_b dbi phi1 &&
no_negative_occurrence_db_b dbi phi2
H0: svar_has_negative_occurrence X phi1
|| svar_has_negative_occurrence X phi2 = false
svar_has_negative_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi phi1 &&
no_positive_occurrence_db_b dbi phi2
H0: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1
|| svar_has_positive_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi phi1 &&
no_negative_occurrence_db_b dbi phi2
H0: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1
|| svar_has_negative_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
*
Σ: Signature
n, dbi: db_index
X: svar
H: if decide (n = dbi) then false else true
H0: false = false
svar_has_positive_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false do 2 case_match; auto; congruence.
*
Σ: Signature
n, dbi: db_index
X: svar
H: true
H0: false = false
svar_has_negative_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false case_match; auto; congruence.
*
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: no_positive_occurrence_db_b dbi phi1 &&
no_positive_occurrence_db_b dbi phi2
H0: svar_has_positive_occurrence X phi1
|| svar_has_positive_occurrence X phi2 = false
svar_has_positive_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: no_positive_occurrence_db_b dbi phi1 = true
H2: no_positive_occurrence_db_b dbi phi2 = true
H0: svar_has_positive_occurrence X phi1
|| svar_has_positive_occurrence X phi2 = false
svar_has_positive_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false apply orb_false_iff in H0 as [H01 H02].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: no_positive_occurrence_db_b dbi phi1 = true
H2: no_positive_occurrence_db_b dbi phi2 = true
H01: svar_has_positive_occurrence X phi1 = false
H02: svar_has_positive_occurrence X phi2 = false
svar_has_positive_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
  erewrite -> (proj1 (IHphi1 _ _)), -> (proj1 (IHphi2 _ _)); auto.
*
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: no_negative_occurrence_db_b dbi phi1 &&
no_negative_occurrence_db_b dbi phi2
H0: svar_has_negative_occurrence X phi1
|| svar_has_negative_occurrence X phi2 = false
svar_has_negative_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b dbi phi1 = true
H2: no_negative_occurrence_db_b dbi phi2 = true
H0: svar_has_negative_occurrence X phi1
|| svar_has_negative_occurrence X phi2 = false
svar_has_negative_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false apply orb_false_iff in H0 as [H01 H02].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b dbi phi1 = true
H2: no_negative_occurrence_db_b dbi phi2 = true
H01: svar_has_negative_occurrence X phi1 = false
H02: svar_has_negative_occurrence X phi2 = false
svar_has_negative_occurrence X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
  erewrite -> (proj2 (IHphi1 _ _)), -> (proj2 (IHphi2 _ _)); auto.
*
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi phi1 &&
no_positive_occurrence_db_b dbi phi2
H0: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1
|| svar_has_positive_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi phi1 = true
H2: no_positive_occurrence_db_b dbi phi2 = true
H0: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1
|| svar_has_positive_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false apply orb_false_iff in H0 as [H01 H02].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi phi1 = true
H2: no_positive_occurrence_db_b dbi phi2 = true
H01: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1 = false
H02: svar_has_positive_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
  erewrite -> (proj2 (IHphi1 _ _)), -> (proj1 (IHphi2 _ _)); auto.
*
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi phi1 &&
no_negative_occurrence_db_b dbi phi2
H0: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1
|| svar_has_negative_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi phi1 = true
H2: no_negative_occurrence_db_b dbi phi2 = true
H0: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1
|| svar_has_negative_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false apply orb_false_iff in H0 as [H01 H02].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi1
   → svar_has_positive_occurrence X phi1 =
     false
     → svar_has_positive_occurrence X
         phi1^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi1
     → svar_has_negative_occurrence X phi1 =
       false
       → svar_has_negative_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar X] = false)
IHphi2: ∀ (dbi : db_index) (X : svar),
  (no_positive_occurrence_db_b dbi phi2
   → svar_has_positive_occurrence X phi2 =
     false
     → svar_has_positive_occurrence X
         phi2^[svar:dbi↦patt_free_svar X] =
       false)
  ∧ (no_negative_occurrence_db_b dbi phi2
     → svar_has_negative_occurrence X phi2 =
       false
       → svar_has_negative_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar X] = false)
dbi: db_index
X: svar
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi phi1 = true
H2: no_negative_occurrence_db_b dbi phi2 = true
H01: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1 = false
H02: svar_has_negative_occurrence X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X
  phi1^[svar:dbi↦patt_free_svar X]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar X] = false
  erewrite -> (proj1 (IHphi1 _ _)), -> (proj2 (IHphi2 _ _)); auto.
Qed.

Lemma positive_negative_occurrence_evar_open : forall (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X (evar_open dbi x ϕ) = false <-> svar_has_positive_occurrence X ϕ = false)
  /\ (svar_has_negative_occurrence X (evar_open dbi x ϕ) = false <-> svar_has_negative_occurrence X ϕ = false).
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X (evar_open dbi x ϕ) =
   false ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X
       (evar_open dbi x ϕ) = false
     ↔ svar_has_negative_occurrence X ϕ = false)
Proof.
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X (evar_open dbi x ϕ) =
   false ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X
       (evar_open dbi x ϕ) = false
     ↔ svar_has_negative_occurrence X ϕ = false)
unfold evar_open.
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X
       ϕ^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ = false)
induction ϕ; intros dbi X; split; simpl; auto; cbn.
Σ: Signature
n: db_index
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  match compare_nat n X with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = false ↔ false = false
Σ: Signature
n: db_index
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  match compare_nat n X with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = false ↔ false = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ svar_has_positive_occurrence dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ svar_has_negative_occurrence dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_negative_occurrence) dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_positive_occurrence) dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X
       ϕ^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
↔ svar_has_positive_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X
       ϕ^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
↔ svar_has_negative_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X
       ϕ^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
↔ svar_has_positive_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ = false)
  ∧ (svar_has_negative_occurrence X
       ϕ^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
↔ svar_has_negative_occurrence dbi ϕ = false
*
Σ: Signature
n: db_index
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  match compare_nat n X with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = false ↔ false = false case_match; auto; congruence.
*
Σ: Signature
n: db_index
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  match compare_nat n X with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = false ↔ false = false case_match; auto; congruence.
*
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ svar_has_positive_occurrence dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false split.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ svar_has_positive_occurrence dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
→ svar_has_positive_occurrence dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_positive_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ svar_has_positive_occurrence dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
svar_has_positive_occurrence dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x] = false
H0: svar_has_positive_occurrence dbi
  ϕ2^[evar:X↦patt_free_evar x] = false
svar_has_positive_occurrence dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
    erewrite -> (proj1 (proj1 (IHϕ1 _ _ _))), -> (proj1 (proj1 (IHϕ2 _ _ _))); eauto.
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
→ svar_has_positive_occurrence dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_positive_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi ϕ1 = false
H0: svar_has_positive_occurrence dbi ϕ2 = false
svar_has_positive_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
    erewrite -> (proj2 (proj1 (IHϕ1 _ _ _))), -> (proj2 (proj1 (IHϕ2 _ _ _))); eauto.
*
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ svar_has_negative_occurrence dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false split.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ svar_has_negative_occurrence dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
→ svar_has_negative_occurrence dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_negative_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ svar_has_negative_occurrence dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
svar_has_negative_occurrence dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x] = false
H0: svar_has_negative_occurrence dbi
  ϕ2^[evar:X↦patt_free_evar x] = false
svar_has_negative_occurrence dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
    erewrite -> (proj1 (proj2 (IHϕ1 _ _ _))), -> (proj1 (proj2 (IHϕ2 _ _ _))); eauto.
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
→ svar_has_negative_occurrence dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_negative_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi ϕ1 = false
H0: svar_has_negative_occurrence dbi ϕ2 = false
svar_has_negative_occurrence dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
    erewrite -> (proj2 (proj2 (IHϕ1 _ _ _))), -> (proj2 (proj2 (IHϕ2 _ _ _))); eauto.
*
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_negative_occurrence) dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false split.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_negative_occurrence) dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_negative_occurrence) dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_positive_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_negative_occurrence) dbi ϕ1
  || svar_has_positive_occurrence dbi ϕ2 = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x] = false
H0: svar_has_positive_occurrence dbi
  ϕ2^[evar:X↦patt_free_evar x] = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
    erewrite -> (proj1 (proj2 (IHϕ1 _ _ _))), -> (proj1 (proj1 (IHϕ2 _ _ _))); eauto.
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_negative_occurrence) dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_positive_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi ϕ1
|| svar_has_positive_occurrence dbi ϕ2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi ϕ1 = false
H0: svar_has_positive_occurrence dbi ϕ2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_positive_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
    erewrite -> (proj2 (proj2 (IHϕ1 _ _ _))), -> (proj2 (proj1 (IHϕ2 _ _ _))); eauto.
*
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
↔ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_positive_occurrence) dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false split.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_positive_occurrence) dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_positive_occurrence) dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_negative_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_positive_occurrence) dbi ϕ1
  || svar_has_negative_occurrence dbi ϕ2 = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x] = false
H0: svar_has_negative_occurrence dbi
  ϕ2^[evar:X↦patt_free_evar x] = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
    erewrite -> (proj1 (proj1 (IHϕ1 _ _ _))), -> (proj1 (proj2 (IHϕ2 _ _ _))); eauto.
  -
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
→ (fix svar_has_positive_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_free_svar X' =>
         if decide (X = X') then true else false
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_positive_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_positive_occurrence X ϕ'
     | _ => false
     end
   with svar_has_negative_occurrence
     (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         svar_has_negative_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         svar_has_positive_occurrence X ϕ₁
         || svar_has_negative_occurrence X ϕ₂
     | patt_exists ϕ' | patt_mu ϕ' =>
         svar_has_negative_occurrence X ϕ'
     | _ => false
     end
   for
   svar_has_positive_occurrence) dbi
    ϕ1^[evar:X↦patt_free_evar x]
  || svar_has_negative_occurrence dbi
       ϕ2^[evar:X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi ϕ1
|| svar_has_negative_occurrence dbi ϕ2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false apply orb_false_iff in H as [? ?].
Σ: Signature
ϕ1, ϕ2: Pattern
IHϕ1: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ1^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ1 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ1^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ1 =
       false)
IHϕ2: ∀ (X : svar) (dbi : db_index) (x : evar),
  (svar_has_positive_occurrence X
     ϕ2^[evar:dbi↦patt_free_evar x] = false
   ↔ svar_has_positive_occurrence X ϕ2 = false)
  ∧ (svar_has_negative_occurrence X
       ϕ2^[evar:dbi↦patt_free_evar x] = false
     ↔ svar_has_negative_occurrence X ϕ2 =
       false)
dbi: svar
X: db_index
x: evar
H: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi ϕ1 = false
H0: svar_has_negative_occurrence dbi ϕ2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) dbi
  ϕ1^[evar:X↦patt_free_evar x]
|| svar_has_negative_occurrence dbi
     ϕ2^[evar:X↦patt_free_evar x] = false
    erewrite -> (proj2 (proj1 (IHϕ1 _ _ _))), -> (proj2 (proj2 (IHϕ2 _ _ _))); eauto.
*
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
↔ svar_has_positive_occurrence dbi ϕ = false split.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
→ svar_has_positive_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi ϕ = false
→ svar_has_positive_occurrence dbi
    ϕ^[evar:S X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
→ svar_has_positive_occurrence dbi ϕ = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
svar_has_positive_occurrence dbi ϕ = false erewrite -> (proj1 (proj1 (IHϕ _ _ _))); eauto.
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi ϕ = false
→ svar_has_positive_occurrence dbi
    ϕ^[evar:S X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi ϕ = false
svar_has_positive_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false erewrite -> (proj2 (proj1 (IHϕ _ _ _))); eauto.
*
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
↔ svar_has_negative_occurrence dbi ϕ = false split.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
→ svar_has_negative_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi ϕ = false
→ svar_has_negative_occurrence dbi
    ϕ^[evar:S X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
→ svar_has_negative_occurrence dbi ϕ = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false
svar_has_negative_occurrence dbi ϕ = false erewrite -> (proj1 (proj2 (IHϕ _ _ _))); eauto.
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi ϕ = false
→ svar_has_negative_occurrence dbi
    ϕ^[evar:S X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi ϕ = false
svar_has_negative_occurrence dbi
  ϕ^[evar:S X↦patt_free_evar x] = false erewrite -> (proj2 (proj2 (IHϕ _ _ _))); eauto.
*
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
↔ svar_has_positive_occurrence dbi ϕ = false split.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
→ svar_has_positive_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi ϕ = false
→ svar_has_positive_occurrence dbi
    ϕ^[evar:X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
→ svar_has_positive_occurrence dbi ϕ = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
svar_has_positive_occurrence dbi ϕ = false erewrite -> (proj1 (proj1 (IHϕ _ _ _))); eauto.
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_positive_occurrence dbi ϕ = false
→ svar_has_positive_occurrence dbi
    ϕ^[evar:X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_positive_occurrence dbi ϕ = false
svar_has_positive_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false erewrite -> (proj2 (proj1 (IHϕ _ _ _))); eauto.
*
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
↔ svar_has_negative_occurrence dbi ϕ = false split.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
→ svar_has_negative_occurrence dbi ϕ = false
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi ϕ = false
→ svar_has_negative_occurrence dbi
    ϕ^[evar:X↦patt_free_evar x] = false
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
→ svar_has_negative_occurrence dbi ϕ = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false
svar_has_negative_occurrence dbi ϕ = false erewrite -> (proj1 (proj2 (IHϕ _ _ _))); eauto.
  -
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
svar_has_negative_occurrence dbi ϕ = false
→ svar_has_negative_occurrence dbi
    ϕ^[evar:X↦patt_free_evar x] = false intro H.
Σ: Signature
ϕ: Pattern
IHϕ: ∀ (X : svar) (dbi : db_index) (x : evar),
(svar_has_positive_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_positive_occurrence X ϕ = false)
∧ (svar_has_negative_occurrence X ϕ^[evar:dbi↦patt_free_evar x] = false
↔ svar_has_negative_occurrence X ϕ = false)
dbi: svar
X: db_index
x: evar
H: svar_has_negative_occurrence dbi ϕ = false
svar_has_negative_occurrence dbi
  ϕ^[evar:X↦patt_free_evar x] = false erewrite -> (proj2 (proj2 (IHϕ _ _ _))); eauto.
Qed.

Corollary positive_occurrence_evar_open : forall (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  svar_has_positive_occurrence X (evar_open dbi x ϕ) = false <-> svar_has_positive_occurrence X ϕ = false.
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  svar_has_positive_occurrence X (evar_open dbi x ϕ) =
  false ↔ svar_has_positive_occurrence X ϕ = false
Proof.
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  svar_has_positive_occurrence X (evar_open dbi x ϕ) =
  false ↔ svar_has_positive_occurrence X ϕ = false
apply positive_negative_occurrence_evar_open.
Qed.

Corollary negative_occurrence_evar_open : forall (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  svar_has_negative_occurrence X (evar_open dbi x ϕ) = false <-> svar_has_negative_occurrence X ϕ = false.
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  svar_has_negative_occurrence X (evar_open dbi x ϕ) =
  false ↔ svar_has_negative_occurrence X ϕ = false
Proof.
Σ: Signature
∀ (ϕ : Pattern) (X : svar) (dbi : db_index) (x : evar),
  svar_has_negative_occurrence X (evar_open dbi x ϕ) =
  false ↔ svar_has_negative_occurrence X ϕ = false
apply positive_negative_occurrence_evar_open.
Qed.

Lemma positive_negative_occurrence_db_svar_open_le : forall (phi : Pattern) (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2 ->
  (
    no_positive_occurrence_db_b dbi1 phi ->
    no_positive_occurrence_db_b dbi1 (svar_open dbi2 X phi)
  )
  /\ (no_negative_occurrence_db_b dbi1 phi -> no_negative_occurrence_db_b dbi1 (svar_open dbi2 X phi)).
Σ: Signature
∀ (phi : Pattern) (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         (svar_open dbi2 X phi))
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           (svar_open dbi2 X phi))
Proof.
Σ: Signature
∀ (phi : Pattern) (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         (svar_open dbi2 X phi))
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           (svar_open dbi2 X phi))
unfold svar_open.
Σ: Signature
∀ (phi : Pattern) (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦patt_free_svar X])
induction phi; intros dbi1 dbi2 X Hneq; split; intros H; simpl in *; auto; cbn in *.
Σ: Signature
n, dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: if decide (n = dbi1) then false else true
no_positive_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end
Σ: Signature
n, dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: true
no_negative_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2
no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2
no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X]
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X]
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b dbi1 phi
no_positive_occurrence_db_b dbi1
  phi^[svar:dbi2↦patt_free_svar X]
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b dbi1 phi
no_negative_occurrence_db_b dbi1
  phi^[svar:dbi2↦patt_free_svar X]
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b (S dbi1) phi
no_positive_occurrence_db_b 
  (S dbi1) phi^[svar:S dbi2↦
  patt_free_svar X]
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b (S dbi1) phi
no_negative_occurrence_db_b 
  (S dbi1) phi^[svar:S dbi2↦
  patt_free_svar X]
+
Σ: Signature
n, dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: if decide (n = dbi1) then false else true
no_positive_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end do 2 case_match; auto; cbn; case_match; auto.
Σ: Signature
n, dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H: true
g: n > dbi2
H1: compare_nat n dbi2 = Nat_greater n dbi2 g
e: Nat.pred n = dbi1
H2: decide (Nat.pred n = dbi1) = left e
false lia.
+
Σ: Signature
n, dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: true
no_negative_occurrence_db_b dbi1
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end case_match; constructor; auto.
+
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2
no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] destruct_and!; split_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_positive_occurrence_db_b dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_positive_occurrence_db_b dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_positive_occurrence_db_b dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true now apply IHphi1.
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_positive_occurrence_db_b dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true now apply IHphi2.
+
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2
no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] destruct_and!; split_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_negative_occurrence_db_b dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_negative_occurrence_db_b dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_negative_occurrence_db_b dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true now apply IHphi1.
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: no_negative_occurrence_db_b dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true now apply IHphi2.
+
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 &&
no_positive_occurrence_db_b dbi1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] destruct_and!; split_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true now apply IHphi1.
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1 = true
H1: no_positive_occurrence_db_b dbi1 phi2 = true
no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true now apply IHphi2.
+
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 &&
no_negative_occurrence_db_b dbi1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] &&
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] destruct_and!; split_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦patt_free_svar X] = true now apply IHphi1.
  -
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi1
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi1
       → no_negative_occurrence_db_b dbi1
           phi1^[svar:dbi2↦
           patt_free_svar X])
IHphi2: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi2
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi2
       → no_negative_occurrence_db_b dbi1
           phi2^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1 = true
H1: no_negative_occurrence_db_b dbi1 phi2 = true
no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦patt_free_svar X] = true now apply IHphi2.
+
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b dbi1 phi
no_positive_occurrence_db_b dbi1
  phi^[svar:dbi2↦patt_free_svar X] now apply IHphi.
+
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b dbi1 phi
no_negative_occurrence_db_b dbi1
  phi^[svar:dbi2↦patt_free_svar X] now apply IHphi.
+
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b (S dbi1) phi
no_positive_occurrence_db_b (S dbi1)
  phi^[svar:S dbi2↦patt_free_svar X] apply IHphi; auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_positive_occurrence_db_b (S dbi1) phi
S dbi1 < S dbi2 lia.
+
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b (S dbi1) phi
no_negative_occurrence_db_b (S dbi1)
  phi^[svar:S dbi2↦patt_free_svar X] apply IHphi; auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi1 dbi2 : db_index) (X : svar),
  dbi1 < dbi2
  → (no_positive_occurrence_db_b dbi1 phi
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦
         patt_free_svar X])
    ∧ (no_negative_occurrence_db_b dbi1 phi
       → no_negative_occurrence_db_b dbi1
           phi^[svar:dbi2↦
           patt_free_svar X])
dbi1, dbi2: db_index
X: svar
Hneq: dbi1 < dbi2
H: no_negative_occurrence_db_b (S dbi1) phi
S dbi1 < S dbi2 lia.
Qed.

Lemma wfp_svar_open : forall (phi : Pattern) (dbi : db_index) (X : svar),
  well_formed_positive phi = true ->
  well_formed_positive (svar_open dbi X phi) = true.
Σ: Signature
∀ (phi : Pattern) (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive (svar_open dbi X phi) = true
Proof.
Σ: Signature
∀ (phi : Pattern) (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive (svar_open dbi X phi) = true
unfold svar_open.
Σ: Signature
∀ (phi : Pattern) (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
induction phi; simpl; intros dbi X H.
Σ: Signature
x: evar
dbi: db_index
X: svar
H: true = true
true = true
Σ: Signature
x: svar
dbi: db_index
X: svar
H: true = true
true = true
Σ: Signature
n, dbi: db_index
X: svar
H: true = true
true = true
Σ: Signature
n, dbi: db_index
X: svar
H: true = true
well_formed_positive
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = true
Σ: Signature
sigma: symbols
dbi: db_index
X: svar
H: true = true
true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true
Σ: Signature
dbi: db_index
X: svar
H: true = true
true = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi = true
well_formed_positive phi^[svar:dbi↦patt_free_svar X] =
true
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] &&
well_formed_positive phi^[svar:S dbi↦patt_free_svar X] =
true
+
Σ: Signature
x: evar
dbi: db_index
X: svar
H: true = true
true = true constructor.
+
Σ: Signature
x: svar
dbi: db_index
X: svar
H: true = true
true = true constructor.
+
Σ: Signature
n, dbi: db_index
X: svar
H: true = true
true = true constructor.
+
Σ: Signature
n, dbi: db_index
X: svar
H: true = true
well_formed_positive
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = true simpl.
Σ: Signature
n, dbi: db_index
X: svar
H: true = true
well_formed_positive
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = true case_match; constructor.
+
Σ: Signature
sigma: symbols
dbi: db_index
X: svar
H: true = true
true = true constructor.
+
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true inversion H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H, H1: well_formed_positive phi1 &&
well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
well_formed_positive phi1 && well_formed_positive phi2 apply andb_true_iff in H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
∧ well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
well_formed_positive phi1 && well_formed_positive phi2 destruct H1 as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
well_formed_positive phi1 && well_formed_positive phi2
  rewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi1 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
well_formed_positive phi1 && well_formed_positive phi2 assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
well_formed_positive phi1 && well_formed_positive phi2 rewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi2 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true && true =
well_formed_positive phi1 && well_formed_positive phi2 assumption.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true && true =
well_formed_positive phi1 && well_formed_positive phi2
  destruct_and!.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H0: well_formed_positive phi1 = true
H3: well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true && true =
well_formed_positive phi1 && well_formed_positive phi2 symmetry.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H0: well_formed_positive phi1 = true
H3: well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi1 && well_formed_positive phi2 =
true && true simpl.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H0: well_formed_positive phi1 = true
H3: well_formed_positive phi2 = true
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi1 && well_formed_positive phi2 =
true split_and!; auto.
+
Σ: Signature
dbi: db_index
X: svar
H: true = true
true = true constructor.
+
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 &&
well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true apply andb_true_iff in H.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi1 = true
∧ well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true destruct H as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi1^[svar:dbi↦patt_free_svar X] &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true rewrite IHphi1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi1 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true apply H1.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true &&
well_formed_positive phi2^[svar:dbi↦patt_free_svar X] =
true rewrite IHphi2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
well_formed_positive phi2 = true
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true && true = true apply H2.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi1 = true
  → well_formed_positive
      phi1^[svar:dbi↦patt_free_svar X] = true
IHphi2: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi2 = true
  → well_formed_positive
      phi2^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: well_formed_positive phi1 = true
H2: well_formed_positive phi2 = true
true && true = true
  reflexivity.
+
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi = true
well_formed_positive phi^[svar:dbi↦patt_free_svar X] =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi = true
well_formed_positive phi^[svar:dbi↦patt_free_svar X] =
true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: well_formed_positive phi = true
well_formed_positive phi^[svar:dbi↦patt_free_svar X] =
true auto.
+
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] &&
well_formed_positive phi^[svar:S dbi↦patt_free_svar X] =
true simpl in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] &&
well_formed_positive phi^[svar:S dbi↦patt_free_svar X] =
true simpl.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] &&
well_formed_positive phi^[svar:S dbi↦patt_free_svar X] =
true apply andb_true_iff in H.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H: no_negative_occurrence_db_b 0 phi = true
∧ well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] &&
well_formed_positive phi^[svar:S dbi↦patt_free_svar X] =
true destruct H as [H1 H2].
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b 0 phi = true
H2: well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] &&
well_formed_positive phi^[svar:S dbi↦patt_free_svar X] =
true
  rewrite IHphi.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b 0 phi = true
H2: well_formed_positive phi = true
well_formed_positive phi = true
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b 0 phi = true
H2: well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] && true = true apply H2.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b 0 phi = true
H2: well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] && true = true rewrite andb_true_r.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b 0 phi = true
H2: well_formed_positive phi = true
no_negative_occurrence_db_b 0
  phi^[svar:S dbi↦patt_free_svar X] = true
  apply positive_negative_occurrence_db_svar_open_le; auto.
Σ: Signature
phi: Pattern
IHphi: ∀ (dbi : db_index) (X : svar),
  well_formed_positive phi = true
  → well_formed_positive
      phi^[svar:dbi↦patt_free_svar X] = true
dbi: db_index
X: svar
H1: no_negative_occurrence_db_b 0 phi = true
H2: well_formed_positive phi = true
0 < S dbi lia.
Qed.

Lemma positive_negative_occurrence_named_svar_open :
forall (phi : Pattern) (X Y : svar) (dbi : db_index),
  X <> Y ->
  (
    svar_has_negative_occurrence X phi = false ->
    svar_has_negative_occurrence X (svar_open dbi Y phi) = false
  ) /\ (
    svar_has_positive_occurrence X phi = false ->
    svar_has_positive_occurrence X (svar_open dbi Y phi) = false
  ).
Σ: Signature
∀ (phi : Pattern) (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi = false
     → svar_has_negative_occurrence X
         (svar_open dbi Y phi) = false)
    ∧ (svar_has_positive_occurrence X phi = false
       → svar_has_positive_occurrence X
           (svar_open dbi Y phi) = false)
Proof.
Σ: Signature
∀ (phi : Pattern) (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi = false
     → svar_has_negative_occurrence X
         (svar_open dbi Y phi) = false)
    ∧ (svar_has_positive_occurrence X phi = false
       → svar_has_positive_occurrence X
           (svar_open dbi Y phi) = false)
unfold svar_open.
Σ: Signature
∀ (phi : Pattern) (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi = false
     → svar_has_negative_occurrence X
         phi^[svar:dbi↦patt_free_svar Y] = false)
    ∧ (svar_has_positive_occurrence X phi = false
       → svar_has_positive_occurrence X
           phi^[svar:dbi↦patt_free_svar Y] = false)
induction phi; intros X Y dbi XneY; split; intros Hneg; simpl in *; auto; try firstorder.
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_negative_occurrence X
  (patt_bound_svar n) = false
svar_has_negative_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar Y
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_bound_svar n) = false
svar_has_positive_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar Y
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_negative_occurrence X
  (patt_app phi1 phi2) = false
svar_has_negative_occurrence X
  (patt_app phi1^[svar:dbi↦
     patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_app phi1 phi2) = false
svar_has_positive_occurrence X
  (patt_app phi1^[svar:dbi↦
     patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_negative_occurrence X
  (patt_imp phi1 phi2) = false
svar_has_negative_occurrence X
  (patt_imp phi1^[svar:dbi↦
     patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_imp phi1 phi2) = false
svar_has_positive_occurrence X
  (patt_imp phi1^[svar:dbi↦
     patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false
-
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_negative_occurrence X
  (patt_bound_svar n) = false
svar_has_negative_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar Y
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false now case_match.
-
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_bound_svar n) = false
svar_has_positive_occurrence X
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar Y
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end = false case_match; try auto.
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_bound_svar n) = false
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
svar_has_positive_occurrence X (patt_free_svar Y) =
false cbn.
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_bound_svar n) = false
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
(if decide (X = Y) then true else false) = false case_match; auto.
Σ: Signature
n: db_index
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_bound_svar n) = false
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
e0: X = Y
H0: decide (X = Y) = left e0
true = false congruence.
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_negative_occurrence X
  (patt_app phi1 phi2) = false
svar_has_negative_occurrence X
  (patt_app phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false apply orb_false_iff in Hneg as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi2 = false
svar_has_negative_occurrence X
  (patt_app phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false
  cbn.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi2 = false
svar_has_negative_occurrence X
  phi1^[svar:dbi↦patt_free_svar Y]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar Y] = false
  now erewrite -> (proj1 (IHphi1 X Y dbi XneY)), -> (proj1 (IHphi2 X Y dbi XneY)).
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_app phi1 phi2) = false
svar_has_positive_occurrence X
  (patt_app phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false apply orb_false_iff in Hneg as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi2 = false
svar_has_positive_occurrence X
  (patt_app phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false
  cbn.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi2 = false
svar_has_positive_occurrence X
  phi1^[svar:dbi↦patt_free_svar Y]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar Y] = false
  now erewrite -> (proj2 (IHphi1 X Y dbi XneY)), -> (proj2 (IHphi2 X Y dbi XneY)).
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_negative_occurrence X
  (patt_imp phi1 phi2) = false
svar_has_negative_occurrence X
  (patt_imp phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false apply orb_false_iff in Hneg as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi2 = false
svar_has_negative_occurrence X
  (patt_imp phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false cbn.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X
  phi1^[svar:dbi↦patt_free_svar Y]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar Y] = false fold svar_has_positive_occurrence.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi2 = false
svar_has_positive_occurrence X
  phi1^[svar:dbi↦patt_free_svar Y]
|| svar_has_negative_occurrence X
     phi2^[svar:dbi↦patt_free_svar Y] = false
  now erewrite -> (proj2 (IHphi1 X Y dbi XneY)), -> (proj1 (IHphi2 X Y dbi XneY)).
-
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
Hneg: svar_has_positive_occurrence X
  (patt_imp phi1 phi2) = false
svar_has_positive_occurrence X
  (patt_imp phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false apply orb_false_iff in Hneg as [H1 H2].
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi2 = false
svar_has_positive_occurrence X
  (patt_imp phi1^[svar:dbi↦patt_free_svar Y]
     phi2^[svar:dbi↦patt_free_svar Y]) = false cbn.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi2 = false
(fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X
  phi1^[svar:dbi↦patt_free_svar Y]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar Y] = false fold svar_has_negative_occurrence.
Σ: Signature
phi1, phi2: Pattern
IHphi1: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi1 =
     false
     → svar_has_negative_occurrence X
         phi1^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi1 =
       false
       → svar_has_positive_occurrence X
           phi1^[svar:dbi↦
           patt_free_svar Y] = false)
IHphi2: ∀ (X Y : svar) (dbi : db_index),
  X ≠ Y
  → (svar_has_negative_occurrence X phi2 =
     false
     → svar_has_negative_occurrence X
         phi2^[svar:dbi↦patt_free_svar Y] =
       false)
    ∧ (svar_has_positive_occurrence X phi2 =
       false
       → svar_has_positive_occurrence X
           phi2^[svar:dbi↦
           patt_free_svar Y] = false)
X, Y: svar
dbi: db_index
XneY: X ≠ Y
H1: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_negative_occurrence) X phi1 = false
H2: (fix svar_has_positive_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_free_svar X' =>
       if decide (X = X') then true else false
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_positive_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_positive_occurrence X ϕ'
   | _ => false
   end
 with svar_has_negative_occurrence
   (X : svar) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       svar_has_negative_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       svar_has_positive_occurrence X ϕ₁
       || svar_has_negative_occurrence X ϕ₂
   | patt_exists ϕ' | patt_mu ϕ' =>
       svar_has_negative_occurrence X ϕ'
   | _ => false
   end
 for
 svar_has_positive_occurrence) X phi2 = false
svar_has_negative_occurrence X
  phi1^[svar:dbi↦patt_free_svar Y]
|| svar_has_positive_occurrence X
     phi2^[svar:dbi↦patt_free_svar Y] = false
  now erewrite -> (proj1 (IHphi1 X Y dbi XneY)), -> (proj2 (IHphi2 X Y dbi XneY)).
Qed.

Corollary evar_open_wfc_aux db1 db2 X phi :
db1 <= db2 ->
well_formed_closed_ex_aux phi db1 ->
evar_open db2 X phi = phi.
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
db1 ≤ db2
→ well_formed_closed_ex_aux phi db1
  → evar_open db2 X phi = phi
Proof.
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
db1 ≤ db2
→ well_formed_closed_ex_aux phi db1
  → evar_open db2 X phi = phi
intros H H0.
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_ex_aux phi db1
evar_open db2 X phi = phi unfold evar_open.
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_ex_aux phi db1
phi^[evar:db2↦patt_free_evar X] = phi eapply well_formed_bevar_subst.
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_ex_aux phi db1
db2 ≥ ?n
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_ex_aux phi db1
well_formed_closed_ex_aux phi ?n 2: eassumption.
Σ: Signature
db1, db2: nat
X: evar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_ex_aux phi db1
db2 ≥ db1 auto.
Qed.

Corollary evar_open_wfc m X phi : well_formed_closed_ex_aux phi 0 -> evar_open m X phi = phi.
Σ: Signature
m: db_index
X: evar
phi: Pattern
well_formed_closed_ex_aux phi 0
→ evar_open m X phi = phi
Proof.
Σ: Signature
m: db_index
X: evar
phi: Pattern
well_formed_closed_ex_aux phi 0
→ evar_open m X phi = phi
intros H.
Σ: Signature
m: db_index
X: evar
phi: Pattern
H: well_formed_closed_ex_aux phi 0
evar_open m X phi = phi
unfold well_formed_closed in H.
Σ: Signature
m: db_index
X: evar
phi: Pattern
H: well_formed_closed_ex_aux phi 0
evar_open m X phi = phi
apply evar_open_wfc_aux with (X := X)(db2 := m) in H.
Σ: Signature
m: db_index
X: evar
phi: Pattern
H: evar_open m X phi = phi
evar_open m X phi = phi
Σ: Signature
m: db_index
X: evar
phi: Pattern
H: well_formed_closed_ex_aux phi 0
0 ≤ m
2: lia.
Σ: Signature
m: db_index
X: evar
phi: Pattern
H: evar_open m X phi = phi
evar_open m X phi = phi
auto.
Qed.

Corollary svar_open_wfc_aux db1 db2 X phi :
db1 <= db2 ->
well_formed_closed_mu_aux phi db1 ->
svar_open db2 X phi = phi.
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
db1 ≤ db2
→ well_formed_closed_mu_aux phi db1
  → svar_open db2 X phi = phi
Proof.
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
db1 ≤ db2
→ well_formed_closed_mu_aux phi db1
  → svar_open db2 X phi = phi
intros H H0.
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_mu_aux phi db1
svar_open db2 X phi = phi unfold evar_open.
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_mu_aux phi db1
svar_open db2 X phi = phi eapply well_formed_bsvar_subst.
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_mu_aux phi db1
db2 ≥ ?k
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_mu_aux phi db1
well_formed_closed_mu_aux phi ?k 2: eassumption.
Σ: Signature
db1, db2: nat
X: svar
phi: Pattern
H: db1 ≤ db2
H0: well_formed_closed_mu_aux phi db1
db2 ≥ db1 auto.
Qed.

Corollary svar_open_wfc m X phi : well_formed_closed_mu_aux phi 0 -> svar_open m X phi = phi.
Σ: Signature
m: db_index
X: svar
phi: Pattern
well_formed_closed_mu_aux phi 0
→ svar_open m X phi = phi
Proof.
Σ: Signature
m: db_index
X: svar
phi: Pattern
well_formed_closed_mu_aux phi 0
→ svar_open m X phi = phi
intros H.
Σ: Signature
m: db_index
X: svar
phi: Pattern
H: well_formed_closed_mu_aux phi 0
svar_open m X phi = phi
unfold well_formed_closed in H.
Σ: Signature
m: db_index
X: svar
phi: Pattern
H: well_formed_closed_mu_aux phi 0
svar_open m X phi = phi
apply svar_open_wfc_aux with (X := X)(db2 := m) in H.
Σ: Signature
m: db_index
X: svar
phi: Pattern
H: svar_open m X phi = phi
svar_open m X phi = phi
Σ: Signature
m: db_index
X: svar
phi: Pattern
H: well_formed_closed_mu_aux phi 0
0 ≤ m
2: lia.
Σ: Signature
m: db_index
X: svar
phi: Pattern
H: svar_open m X phi = phi
svar_open m X phi = phi
auto.
Qed.

Corollary evar_open_bsvar_subst m phi1 phi2 dbi X
: well_formed_closed phi2 ->
  evar_open m X (bsvar_subst phi1 phi2 dbi)
  = bsvar_subst (evar_open m X phi1) phi2 dbi.
Σ: Signature
m: db_index
phi1, phi2: Pattern
dbi: db_index
X: evar
well_formed_closed phi2
→ evar_open m X phi1^[svar:dbi↦phi2] =
  (evar_open m X phi1)^[svar:dbi↦phi2]
Proof.
Σ: Signature
m: db_index
phi1, phi2: Pattern
dbi: db_index
X: evar
well_formed_closed phi2
→ evar_open m X phi1^[svar:dbi↦phi2] =
  (evar_open m X phi1)^[svar:dbi↦phi2]
intro H.
Σ: Signature
m: db_index
phi1, phi2: Pattern
dbi: db_index
X: evar
H: well_formed_closed phi2
evar_open m X phi1^[svar:dbi↦phi2] =
(evar_open m X phi1)^[svar:dbi↦phi2] apply bevar_subst_bsvar_subst; auto.
Qed.

Corollary svar_open_bevar_subst m phi1 phi2 dbi X
: well_formed_closed phi2 ->
  svar_open m X (bevar_subst phi1 phi2 dbi)
  = bevar_subst (svar_open m X phi1) phi2 dbi.
Σ: Signature
m: db_index
phi1, phi2: Pattern
dbi: db_index
X: svar
well_formed_closed phi2
→ svar_open m X phi1^[evar:dbi↦phi2] =
  (svar_open m X phi1)^[evar:dbi↦phi2]
Proof.
Σ: Signature
m: db_index
phi1, phi2: Pattern
dbi: db_index
X: svar
well_formed_closed phi2
→ svar_open m X phi1^[evar:dbi↦phi2] =
  (svar_open m X phi1)^[evar:dbi↦phi2]
intro H.
Σ: Signature
m: db_index
phi1, phi2: Pattern
dbi: db_index
X: svar
H: well_formed_closed phi2
svar_open m X phi1^[evar:dbi↦phi2] =
(svar_open m X phi1)^[evar:dbi↦phi2] apply eq_sym, bevar_subst_bsvar_subst; auto.
Qed.

Corollary svar_open_bsvar_subst_higher m phi1 phi2 dbi X
: well_formed_closed phi2 ->
  m < dbi ->
  svar_open m X (bsvar_subst phi1 phi2 dbi)
  = bsvar_subst (svar_open m X phi1) phi2 (pred dbi).
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
well_formed_closed phi2
→ m < dbi
  → svar_open m X phi1^[svar:dbi↦phi2] =
    (svar_open m X phi1)^[svar:Init.Nat.pred dbi↦phi2]
Proof.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
well_formed_closed phi2
→ m < dbi
  → svar_open m X phi1^[svar:dbi↦phi2] =
    (svar_open m X phi1)^[svar:Init.Nat.pred dbi↦phi2]
intros H H0.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H: well_formed_closed phi2
H0: m < dbi
svar_open m X phi1^[svar:dbi↦phi2] =
(svar_open m X phi1)^[svar:Init.Nat.pred dbi↦phi2] apply bsvar_subst_comm_higher; auto.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H: well_formed_closed phi2
H0: m < dbi
well_formed_closed_mu_aux phi2 0
unfold well_formed_closed in *.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H: well_formed_closed_mu_aux phi2 0 &&
well_formed_closed_ex_aux phi2 0
H0: m < dbi
well_formed_closed_mu_aux phi2 0 destruct_and!.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H1: well_formed_closed_mu_aux phi2 0 = true
H2: well_formed_closed_ex_aux phi2 0 = true
H0: m < dbi
well_formed_closed_mu_aux phi2 0 auto.
Qed.

Corollary svar_open_bsvar_subst_lower m phi1 phi2 dbi X
: well_formed_closed phi2 ->
  m > dbi ->
  svar_open m X (bsvar_subst phi1 phi2 dbi)
  = bsvar_subst (svar_open (S m) X phi1) phi2 dbi.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
well_formed_closed phi2
→ m > dbi
  → svar_open m X phi1^[svar:dbi↦phi2] =
    (svar_open (S m) X phi1)^[svar:dbi↦phi2]
Proof.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
well_formed_closed phi2
→ m > dbi
  → svar_open m X phi1^[svar:dbi↦phi2] =
    (svar_open (S m) X phi1)^[svar:dbi↦phi2]
intros H H0.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H: well_formed_closed phi2
H0: m > dbi
svar_open m X phi1^[svar:dbi↦phi2] =
(svar_open (S m) X phi1)^[svar:dbi↦phi2] apply bsvar_subst_comm_lower; auto.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H: well_formed_closed phi2
H0: m > dbi
well_formed_closed_mu_aux phi2 0
unfold well_formed_closed in *.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H: well_formed_closed_mu_aux phi2 0 &&
well_formed_closed_ex_aux phi2 0
H0: m > dbi
well_formed_closed_mu_aux phi2 0 destruct_and!.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: svar
H1: well_formed_closed_mu_aux phi2 0 = true
H2: well_formed_closed_ex_aux phi2 0 = true
H0: m > dbi
well_formed_closed_mu_aux phi2 0 auto.
Qed.

Corollary evar_open_bevar_subst_higher m phi1 phi2 dbi X
: well_formed_closed_ex_aux phi2 0 ->
  m < dbi ->
  evar_open m X (bevar_subst phi1 phi2 dbi)
  = bevar_subst (evar_open m X phi1) phi2 (pred dbi).
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
well_formed_closed_ex_aux phi2 0
→ m < dbi
  → evar_open m X phi1^[evar:dbi↦phi2] =
    (evar_open m X phi1)^[evar:Init.Nat.pred dbi↦phi2]
Proof.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
well_formed_closed_ex_aux phi2 0
→ m < dbi
  → evar_open m X phi1^[evar:dbi↦phi2] =
    (evar_open m X phi1)^[evar:Init.Nat.pred dbi↦phi2]
intros H H0.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
H: well_formed_closed_ex_aux phi2 0
H0: m < dbi
evar_open m X phi1^[evar:dbi↦phi2] =
(evar_open m X phi1)^[evar:Init.Nat.pred dbi↦phi2] apply bevar_subst_comm_higher; auto.
Qed.

Corollary evar_open_bevar_subst_lower m phi1 phi2 dbi X
: well_formed_closed phi2 ->
  m > dbi ->
  evar_open m X (bevar_subst phi1 phi2 dbi)
  = bevar_subst (evar_open (S m) X phi1) phi2 dbi.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
well_formed_closed phi2
→ m > dbi
  → evar_open m X phi1^[evar:dbi↦phi2] =
    (evar_open (S m) X phi1)^[evar:dbi↦phi2]
Proof.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
well_formed_closed phi2
→ m > dbi
  → evar_open m X phi1^[evar:dbi↦phi2] =
    (evar_open (S m) X phi1)^[evar:dbi↦phi2]
intros H H0.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
H: well_formed_closed phi2
H0: m > dbi
evar_open m X phi1^[evar:dbi↦phi2] =
(evar_open (S m) X phi1)^[evar:dbi↦phi2] apply bevar_subst_comm_lower; auto.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
H: well_formed_closed phi2
H0: m > dbi
well_formed_closed_ex_aux phi2 0
unfold well_formed_closed in *.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
H: well_formed_closed_mu_aux phi2 0 &&
well_formed_closed_ex_aux phi2 0
H0: m > dbi
well_formed_closed_ex_aux phi2 0 destruct_and!.
Σ: Signature
m: nat
phi1, phi2: Pattern
dbi: nat
X: evar
H1: well_formed_closed_mu_aux phi2 0 = true
H2: well_formed_closed_ex_aux phi2 0 = true
H0: m > dbi
well_formed_closed_ex_aux phi2 0 auto.
Qed.



Lemma free_svars_bsvar_subst' :
forall φ ψ dbi X,
  (X ∈ free_svars (bsvar_subst φ ψ dbi)) <->
  ((X ∈ (free_svars ψ) /\ bsvar_occur φ dbi) \/ (X ∈ (free_svars φ))).
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
    ∨ X ∈ free_svars φ
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
    ∨ X ∈ free_svars φ
induction φ; intros ψ dbi X; simpl.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: svar
X ∈ {[x]} ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ {[x]}
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
X
∈ free_svars
    match compare_nat n dbi with
    | Nat_less _ _ _ => patt_bound_svar n
    | Nat_equal _ _ _ => ψ
    | Nat_greater _ _ _ =>
        patt_bound_svar (Nat.pred n)
    end
↔ X ∈ free_svars ψ
  ∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X
∈ free_svars φ1^[svar:dbi↦ψ]
  ∪ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∪ free_svars φ2
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X
∈ free_svars φ1^[svar:dbi↦ψ]
  ∪ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∪ free_svars φ2
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
    ∨ X ∈ free_svars φ
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
  ∨ X ∈ free_svars φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
    ∨ X ∈ free_svars φ
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ^[svar:S dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ (S dbi)
  ∨ X ∈ free_svars φ
-
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
  destruct H.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false
X ∈ ∅
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ ∅
X ∈ ∅
  destruct H.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: false
X ∈ ∅
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ ∅
X ∈ ∅ congruence.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ ∅
X ∈ ∅ assumption.
-
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: svar
X ∈ {[x]} ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ {[x]} split; intros H; auto.
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false ∨ X ∈ {[x]}
X ∈ {[x]}
  destruct H; auto.
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false
X ∈ {[x]}
  destruct H; congruence.
-
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
  destruct H; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false
X ∈ ∅
  destruct H; congruence.
-
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
X
∈ free_svars
    match compare_nat n dbi with
    | Nat_less _ _ _ => patt_bound_svar n
    | Nat_equal _ _ _ => ψ
    | Nat_greater _ _ _ =>
        patt_bound_svar (Nat.pred n)
    end
↔ X ∈ free_svars ψ
  ∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ case_match; split; intros H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_svars (patt_bound_svar n)
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars (patt_bound_svar n)
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars ψ
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars (patt_bound_svar (Nat.pred n))
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars (patt_bound_svar (Nat.pred n))
  +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_svars (patt_bound_svar n)
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ simpl in H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ ∅
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ set_solver.
  +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars (patt_bound_svar n) destruct H' as [H'|H'].
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_svars (patt_bound_svar n)
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ ∅
X ∈ free_svars (patt_bound_svar n)
    *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_svars (patt_bound_svar n) destruct H'; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H0: X ∈ free_svars ψ
H1: if decide (n = dbi) then true else false
X ∈ free_svars (patt_bound_svar n) case_match; auto; subst.
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
l: dbi < dbi
H: compare_nat dbi dbi = Nat_less dbi dbi l
H0: X ∈ free_svars ψ
H2: decide (dbi = dbi) = left (erefl dbi)
H1: true
X ∈ free_svars (patt_bound_svar dbi)
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H0: X ∈ free_svars ψ
n0: n ≠ dbi
H2: decide (n = dbi) = right n0
H1: false
X ∈ free_svars (patt_bound_svar n) lia.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H0: X ∈ free_svars ψ
n0: n ≠ dbi
H2: decide (n = dbi) = right n0
H1: false
X ∈ free_svars (patt_bound_svar n) congruence.
    *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ ∅
X ∈ free_svars (patt_bound_svar n) set_solver.
  +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ left.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) split; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
if decide (n = dbi) then true else false case_match; auto.
  +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars ψ simpl in H.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars ψ set_solver.
  +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars (patt_bound_svar (Nat.pred n))
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ simpl in H.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars (patt_bound_svar (Nat.pred n))
X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ set_solver.
  +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_svars (patt_bound_svar (Nat.pred n)) destruct H' as [H'|H'].
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_svars (patt_bound_svar (Nat.pred n))
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ ∅
X ∈ free_svars (patt_bound_svar (Nat.pred n))
    *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_svars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_svars (patt_bound_svar (Nat.pred n)) destruct H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H0: X ∈ free_svars ψ
H1: if decide (n = dbi) then true else false
X ∈ free_svars (patt_bound_svar (Nat.pred n)) case_match; try lia; congruence.
    *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: svar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ ∅
X ∈ free_svars (patt_bound_svar (Nat.pred n)) set_solver.
-
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅ split; intros H'; auto.
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
H': X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
  destruct H' as [H'|H'].
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
H': X ∈ free_svars ψ ∧ false
X ∈ ∅
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
H': X ∈ ∅
X ∈ ∅
  +
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
H': X ∈ free_svars ψ ∧ false
X ∈ ∅ destruct H'.
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: false
X ∈ ∅ congruence.
  +
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: svar
H': X ∈ ∅
X ∈ ∅ set_solver.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X
∈ free_svars φ1^[svar:dbi↦ψ]
  ∪ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∪ free_svars φ2 rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ1^[svar:dbi↦ψ]
∨ X ∈ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∪ free_svars φ2
  rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ1^[svar:dbi↦ψ]
∨ X ∈ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
  rewrite IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1) ∨ X ∈ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
  rewrite IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
  split; intros H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: (X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: (X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
         split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ1 dbi
bsvar_occur φ1 dbi || bsvar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ1 dbi
true || bsvar_occur φ2 dbi auto.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars φ1 assumption.
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
         split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ2 dbi
bsvar_occur φ1 dbi || bsvar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ2 dbi
bsvar_occur φ1 dbi || true apply orbT.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars φ2 assumption.
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H as [H1 H2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
      destruct (decide (bsvar_occur φ1 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi split; assumption.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct (decide (bsvar_occur φ2 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
         2: {
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 apply orb_prop in H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi = true
∨ bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ1 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
              rewrite H in n.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ1 dbi = true
n0: ¬ bsvar_occur φ2 dbi
n: ¬ true
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 congruence.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
              rewrite H in n0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ true
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 congruence.
         }
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
         right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2
         left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi split; assumption.
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars φ1 assumption.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars φ2 assumption.
-
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
X ∈ ∅ ↔ X ∈ free_svars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
  destruct H.
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false
X ∈ ∅
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ ∅
X ∈ ∅
  +
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ false
X ∈ ∅ destruct H.
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: false
X ∈ ∅ congruence.
  +
Σ: Signature
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ ∅
X ∈ ∅ set_solver.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X
∈ free_svars φ1^[svar:dbi↦ψ]
  ∪ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∪ free_svars φ2 rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ1^[svar:dbi↦ψ]
∨ X ∈ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∪ free_svars φ2
  rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ1^[svar:dbi↦ψ]
∨ X ∈ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
  rewrite IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1) ∨ X ∈ free_svars φ2^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
  rewrite IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
↔ X ∈ free_svars ψ
  ∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
  split; intros H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: (X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: (X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ1 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
         split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ1 dbi
bsvar_occur φ1 dbi || bsvar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ1 dbi
true || bsvar_occur φ2 dbi auto.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars φ1 assumption.
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ2 dbi
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
         split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ2 dbi
bsvar_occur φ1 dbi || bsvar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
H0: bsvar_occur φ2 dbi
bsvar_occur φ1 dbi || true apply orbT.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars φ1 ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars φ2 assumption.
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
∨ X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars ψ
∧ bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H as [H1 H2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
      destruct (decide (bsvar_occur φ1 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
i: bsvar_occur φ1 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi split; assumption.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct (decide (bsvar_occur φ2 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
         2: {
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 apply orb_prop in H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi = true
∨ bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ1 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
              rewrite H in n.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ1 dbi = true
n0: ¬ bsvar_occur φ2 dbi
n: ¬ true
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 congruence.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
              rewrite H in n0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H: bsvar_occur φ2 dbi = true
n: ¬ bsvar_occur φ1 dbi
n0: ¬ true
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 congruence.
         }
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
         right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2
         left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H1: X ∈ free_svars ψ
H2: bsvar_occur φ1 dbi || bsvar_occur φ2 dbi
n: ¬ bsvar_occur φ1 dbi
i: bsvar_occur φ2 dbi
X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi split; assumption.
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1 ∨ X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
∨ X ∈ free_svars φ1 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ1
X ∈ free_svars φ1 assumption.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
(X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
 ∨ X ∈ free_svars φ1)
∨ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
  ∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
∨ X ∈ free_svars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ1^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ1 dbi
    ∨ X ∈ free_svars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
  X ∈ free_svars φ2^[svar:dbi↦ψ]
  ↔ X ∈ free_svars ψ ∧ bsvar_occur φ2 dbi
    ∨ X ∈ free_svars φ2
ψ: Pattern
dbi: db_index
X: svar
H: X ∈ free_svars φ2
X ∈ free_svars φ2 assumption.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
X ∈ free_svars φ^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
  ∨ X ∈ free_svars φ rewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
X ∈ free_svars φ^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars ψ ∧ bsvar_occur φ dbi
∨ X ∈ free_svars φ
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi
  ∨ X ∈ free_svars φ auto.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
X ∈ free_svars φ^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars φ^[svar:S dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ (S dbi)
  ∨ X ∈ free_svars φ rewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : svar),
X ∈ free_svars φ^[svar:dbi↦ψ]
↔ X ∈ free_svars ψ ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
ψ: Pattern
dbi: db_index
X: svar
X ∈ free_svars ψ ∧ bsvar_occur φ (S dbi)
∨ X ∈ free_svars φ
↔ X ∈ free_svars ψ ∧ bsvar_occur φ (S dbi)
  ∨ X ∈ free_svars φ auto.
Qed.

Lemma free_evars_bevar_subst' :
forall φ ψ dbi X,
  (X ∈ free_evars (bevar_subst φ ψ dbi)) <->
  ((X ∈ (free_evars ψ) /\ bevar_occur φ dbi) \/ (X ∈ (free_evars φ))).
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
    ∨ X ∈ free_evars φ
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
    ∨ X ∈ free_evars φ
induction φ; intros ψ dbi X; simpl.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
X ∈ {[x]} ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ {[x]}
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
X
∈ free_evars
    match compare_nat n dbi with
    | Nat_less _ _ _ => patt_bound_evar n
    | Nat_equal _ _ _ => ψ
    | Nat_greater _ _ _ =>
        patt_bound_evar (Nat.pred n)
    end
↔ X ∈ free_evars ψ
  ∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X
∈ free_evars φ1^[evar:dbi↦ψ]
  ∪ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∪ free_evars φ2
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X
∈ free_evars φ1^[evar:dbi↦ψ]
  ∪ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∪ free_evars φ2
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
    ∨ X ∈ free_evars φ
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ^[evar:S dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ (S dbi)
  ∨ X ∈ free_evars φ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
    ∨ X ∈ free_evars φ
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
  ∨ X ∈ free_evars φ
-
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
X ∈ {[x]} ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ {[x]} split; intros H; auto.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false ∨ X ∈ {[x]}
X ∈ {[x]}
  destruct H.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ {[x]}
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ {[x]}
X ∈ {[x]}
  destruct H.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: false
X ∈ {[x]}
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ {[x]}
X ∈ {[x]} congruence.
Σ: Signature
x: evar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ {[x]}
X ∈ {[x]} assumption.
-
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
  destruct H; auto.
Σ: Signature
x: svar
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ ∅
  destruct H; congruence.
-
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
X
∈ free_evars
    match compare_nat n dbi with
    | Nat_less _ _ _ => patt_bound_evar n
    | Nat_equal _ _ _ => ψ
    | Nat_greater _ _ _ =>
        patt_bound_evar (Nat.pred n)
    end
↔ X ∈ free_evars ψ
  ∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ case_match; split; intros H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_evars (patt_bound_evar n)
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars (patt_bound_evar n)
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars ψ
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_evars (patt_bound_evar (Nat.pred n))
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars (patt_bound_evar (Nat.pred n))
  +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_evars (patt_bound_evar n)
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ simpl in H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ ∅
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ set_solver.
  +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars (patt_bound_evar n) destruct H' as [H'|H'].
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_evars (patt_bound_evar n)
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ ∅
X ∈ free_evars (patt_bound_evar n)
    *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_evars (patt_bound_evar n) destruct H'; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H0: X ∈ free_evars ψ
H1: if decide (n = dbi) then true else false
X ∈ free_evars (patt_bound_evar n) case_match; auto; subst.
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
l: dbi < dbi
H: compare_nat dbi dbi = Nat_less dbi dbi l
H0: X ∈ free_evars ψ
H2: decide (dbi = dbi) = left (erefl dbi)
H1: true
X ∈ free_evars (patt_bound_evar dbi)
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H0: X ∈ free_evars ψ
n0: n ≠ dbi
H2: decide (n = dbi) = right n0
H1: false
X ∈ free_evars (patt_bound_evar n) lia.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H0: X ∈ free_evars ψ
n0: n ≠ dbi
H2: decide (n = dbi) = right n0
H1: false
X ∈ free_evars (patt_bound_evar n) congruence.
    *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
l: n < dbi
H: compare_nat n dbi = Nat_less n dbi l
H': X ∈ ∅
X ∈ free_evars (patt_bound_evar n) set_solver.
  +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ left.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) split; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
if decide (n = dbi) then true else false case_match; auto.
  +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars ψ simpl in H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
e: n = dbi
H: compare_nat n dbi = Nat_equal n dbi e
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars ψ set_solver.
  +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_evars (patt_bound_evar (Nat.pred n))
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ simpl in H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ ∅
X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false) ∨ X ∈ ∅ set_solver.
  +
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
∨ X ∈ ∅
X ∈ free_evars (patt_bound_evar (Nat.pred n)) destruct H' as [H'|H'].
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_evars (patt_bound_evar (Nat.pred n))
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ ∅
X ∈ free_evars (patt_bound_evar (Nat.pred n))
    *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ free_evars ψ
∧ (if decide (n = dbi) then true else false)
X ∈ free_evars (patt_bound_evar (Nat.pred n)) destruct H'.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H0: X ∈ free_evars ψ
H1: if decide (n = dbi) then true else false
X ∈ free_evars (patt_bound_evar (Nat.pred n)) case_match; try lia; congruence.
    *
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H': X ∈ ∅
X ∈ free_evars (patt_bound_evar (Nat.pred n)) set_solver.
-
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
  destruct H; auto.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ ∅
  destruct H; congruence.
-
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
  destruct H.
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ ∅
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ ∅
X ∈ ∅
  +
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ ∅ destruct H.
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: false
X ∈ ∅ congruence.
  +
Σ: Signature
sigma: symbols
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ ∅
X ∈ ∅ set_solver.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X
∈ free_evars φ1^[evar:dbi↦ψ]
  ∪ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∪ free_evars φ2 rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ1^[evar:dbi↦ψ]
∨ X ∈ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∪ free_evars φ2
  rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ1^[evar:dbi↦ψ]
∨ X ∈ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
  rewrite IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1) ∨ X ∈ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
  rewrite IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
  split; intros H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: (X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: (X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
         split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ1 dbi
bevar_occur φ1 dbi || bevar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ1 dbi
true || bevar_occur φ2 dbi auto.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars φ1 assumption.
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
         split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ2 dbi
bevar_occur φ1 dbi || bevar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ2 dbi
bevar_occur φ1 dbi || true apply orbT.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars φ2 assumption.
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H as [H1 H2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
      destruct (decide (bevar_occur φ1 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
X ∈ free_evars ψ ∧ bevar_occur φ1 dbi split; assumption.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct (decide (bevar_occur φ2 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
         2: {
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 apply orb_prop in H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi = true
∨ bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ1 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
              rewrite H in n.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ1 dbi = true
n0: ¬ bevar_occur φ2 dbi
n: ¬ true
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 congruence.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
              rewrite H in n0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ true
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 congruence.
         }
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
         right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2
         left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
X ∈ free_evars ψ ∧ bevar_occur φ2 dbi split; assumption.
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars φ1 assumption.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars φ2 assumption.
-
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
X ∈ ∅ ↔ X ∈ free_evars ψ ∧ false ∨ X ∈ ∅ split; intros H; auto.
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false ∨ X ∈ ∅
X ∈ ∅
  destruct H.
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ ∅
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ ∅
X ∈ ∅
  +
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ false
X ∈ ∅ destruct H.
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: false
X ∈ ∅ congruence.
  +
Σ: Signature
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ ∅
X ∈ ∅ set_solver.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X
∈ free_evars φ1^[evar:dbi↦ψ]
  ∪ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∪ free_evars φ2 rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ1^[evar:dbi↦ψ]
∨ X ∈ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∪ free_evars φ2
  rewrite elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ1^[evar:dbi↦ψ]
∨ X ∈ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
  rewrite IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1) ∨ X ∈ free_evars φ2^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
  rewrite IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
↔ X ∈ free_evars ψ
  ∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
  split; intros H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: (X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: (X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ1 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
         split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ1 dbi
bevar_occur φ1 dbi || bevar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ1 dbi
true || bevar_occur φ2 dbi auto.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars φ1 assumption.
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ2 dbi
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
         split; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ2 dbi
bevar_occur φ1 dbi || bevar_occur φ2 dbi rewrite H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
H0: bevar_occur φ2 dbi
bevar_occur φ1 dbi || true apply orbT.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars φ1 ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars φ2 assumption.
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
∨ X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars ψ
∧ bevar_occur φ1 dbi || bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H as [H1 H2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
      destruct (decide (bevar_occur φ1 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
i: bevar_occur φ1 dbi
X ∈ free_evars ψ ∧ bevar_occur φ1 dbi split; assumption.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct (decide (bevar_occur φ2 dbi)).
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
         2: {
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 apply orb_prop in H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi = true
∨ bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ1 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
              rewrite H in n.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ1 dbi = true
n0: ¬ bevar_occur φ2 dbi
n: ¬ true
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 congruence.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
              rewrite H in n0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H: bevar_occur φ2 dbi = true
n: ¬ bevar_occur φ1 dbi
n0: ¬ true
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 congruence.
         }
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
         right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2
         left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H1: X ∈ free_evars ψ
H2: bevar_occur φ1 dbi || bevar_occur φ2 dbi
n: ¬ bevar_occur φ1 dbi
i: bevar_occur φ2 dbi
X ∈ free_evars ψ ∧ bevar_occur φ2 dbi split; assumption.
    *
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1 ∨ X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 destruct H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 left.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
∨ X ∈ free_evars φ1 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ1
X ∈ free_evars φ1 assumption.
      --
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
(X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
 ∨ X ∈ free_evars φ1)
∨ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
  ∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
∨ X ∈ free_evars φ2 right.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ1^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ1 dbi
    ∨ X ∈ free_evars φ1
IHφ2: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
  X ∈ free_evars φ2^[evar:dbi↦ψ]
  ↔ X ∈ free_evars ψ ∧ bevar_occur φ2 dbi
    ∨ X ∈ free_evars φ2
ψ: Pattern
dbi: db_index
X: evar
H: X ∈ free_evars φ2
X ∈ free_evars φ2 assumption.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
X ∈ free_evars φ^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi ∨ X ∈ free_evars φ
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ^[evar:S dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ (S dbi)
  ∨ X ∈ free_evars φ rewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
X ∈ free_evars φ^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi ∨ X ∈ free_evars φ
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars ψ ∧ bevar_occur φ (S dbi)
∨ X ∈ free_evars φ
↔ X ∈ free_evars ψ ∧ bevar_occur φ (S dbi)
  ∨ X ∈ free_evars φ auto.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
X ∈ free_evars φ^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi ∨ X ∈ free_evars φ
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars φ^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
  ∨ X ∈ free_evars φ rewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (dbi : db_index) (X : evar),
X ∈ free_evars φ^[evar:dbi↦ψ]
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi ∨ X ∈ free_evars φ
ψ: Pattern
dbi: db_index
X: evar
X ∈ free_evars ψ ∧ bevar_occur φ dbi
∨ X ∈ free_evars φ
↔ X ∈ free_evars ψ ∧ bevar_occur φ dbi
  ∨ X ∈ free_evars φ auto.
Qed.


Lemma free_svars_bsvar_subst :
forall φ ψ dbi,
free_svars (bsvar_subst φ ψ dbi) ⊆ union (free_svars ψ) (free_svars φ).
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index),
  free_svars φ^[svar:dbi↦ψ]
  ⊆ free_svars ψ ∪ free_svars φ
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index),
  free_svars φ^[svar:dbi↦ψ]
  ⊆ free_svars ψ ∪ free_svars φ
induction φ; intros ψ dbi; simpl; try set_solver.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
free_svars
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end ⊆ free_svars ψ ∪ ∅
case_match; simpl; set_solver.
Qed.

Lemma free_evars_bevar_subst :
forall φ ψ dbi,
free_evars (bevar_subst φ ψ dbi) ⊆ union (free_evars ψ) (free_evars φ).
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index),
  free_evars φ^[evar:dbi↦ψ]
  ⊆ free_evars ψ ∪ free_evars φ
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (dbi : db_index),
  free_evars φ^[evar:dbi↦ψ]
  ⊆ free_evars ψ ∪ free_evars φ
induction φ; intros ψ dbi Hwf; simpl; try set_solver.
Σ: Signature
n: db_index
ψ: Pattern
dbi: db_index
Hwf: evar
Hwf
∈ free_evars
    match compare_nat n dbi with
    | Nat_less _ _ _ => patt_bound_evar n
    | Nat_equal _ _ _ => ψ
    | Nat_greater _ _ _ =>
        patt_bound_evar (Nat.pred n)
    end → Hwf ∈ free_evars ψ ∪ ∅
case_match; simpl; set_solver.
Qed.

Lemma free_svars_svar_open'' :
forall φ dbi X Y,
  (X ∈ free_svars (svar_open dbi Y φ)) <->
  (((X = Y) /\ (bsvar_occur φ dbi)) \/ (X ∈ (free_svars φ))).
Σ: Signature
∀ (φ : Pattern) (dbi : db_index) (X Y : svar),
  X ∈ free_svars (svar_open dbi Y φ)
  ↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
Proof.
Σ: Signature
∀ (φ : Pattern) (dbi : db_index) (X Y : svar),
  X ∈ free_svars (svar_open dbi Y φ)
  ↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
intros φ dbi X Y.
Σ: Signature
φ: Pattern
dbi: db_index
X, Y: svar
X ∈ free_svars (svar_open dbi Y φ)
↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
unfold svar_open.
Σ: Signature
φ: Pattern
dbi: db_index
X, Y: svar
X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
pose proof (Htmp := free_svars_bsvar_subst' φ (patt_free_svar Y) dbi X).
Σ: Signature
φ: Pattern
dbi: db_index
X, Y: svar
Htmp: X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X ∈ free_svars (patt_free_svar Y)
  ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
simpl in Htmp.
Σ: Signature
φ: Pattern
dbi: db_index
X, Y: svar
Htmp: X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X ∈ {[Y]} ∧ bsvar_occur φ dbi
  ∨ X ∈ free_svars φ
X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
assert (X ∈ @singleton _ SVarSet _ Y <-> X = Y) by set_solver.
Σ: Signature
φ: Pattern
dbi: db_index
X, Y: svar
Htmp: X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X ∈ {[Y]} ∧ bsvar_occur φ dbi
  ∨ X ∈ free_svars φ
H: X ∈ {[Y]} ↔ X = Y
X ∈ free_svars φ^[svar:dbi↦patt_free_svar Y]
↔ X = Y ∧ bsvar_occur φ dbi ∨ X ∈ free_svars φ
tauto.
Qed.

Corollary free_svars_svar_open ϕ X dbi :
free_svars (svar_open dbi X ϕ) ⊆ union (singleton X) (free_svars ϕ).
Σ: Signature
ϕ: Pattern
X: svar
dbi: db_index
free_svars (svar_open dbi X ϕ) ⊆ {[X]} ∪ free_svars ϕ
Proof.
Σ: Signature
ϕ: Pattern
X: svar
dbi: db_index
free_svars (svar_open dbi X ϕ) ⊆ {[X]} ∪ free_svars ϕ
apply free_svars_bsvar_subst; auto.
Qed.

Lemma free_evars_evar_open'' :
forall φ dbi x y,
  (x ∈ free_evars (evar_open dbi y φ)) <->
  ((x = y /\ bevar_occur φ dbi) \/ (x ∈ (free_evars φ))).
Σ: Signature
∀ (φ : Pattern) (dbi : db_index) (x y : evar),
  x ∈ free_evars (evar_open dbi y φ)
  ↔ x = y ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
Proof.
Σ: Signature
∀ (φ : Pattern) (dbi : db_index) (x y : evar),
  x ∈ free_evars (evar_open dbi y φ)
  ↔ x = y ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
intros φ dbi x y.
Σ: Signature
φ: Pattern
dbi: db_index
x, y: evar
x ∈ free_evars (evar_open dbi y φ)
↔ x = y ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
unfold evar_open.
Σ: Signature
φ: Pattern
dbi: db_index
x, y: evar
x ∈ free_evars φ^[evar:dbi↦patt_free_evar y]
↔ x = y ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
pose proof (Htmp := free_evars_bevar_subst' φ (patt_free_evar y) dbi x).
Σ: Signature
φ: Pattern
dbi: db_index
x, y: evar
Htmp: x ∈ free_evars φ^[evar:dbi↦patt_free_evar y]
↔ x ∈ free_evars (patt_free_evar y)
  ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
x ∈ free_evars φ^[evar:dbi↦patt_free_evar y]
↔ x = y ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
simpl in Htmp.
Σ: Signature
φ: Pattern
dbi: db_index
x, y: evar
Htmp: x ∈ free_evars φ^[evar:dbi↦patt_free_evar y]
↔ x ∈ {[y]} ∧ bevar_occur φ dbi
  ∨ x ∈ free_evars φ
x ∈ free_evars φ^[evar:dbi↦patt_free_evar y]
↔ x = y ∧ bevar_occur φ dbi ∨ x ∈ free_evars φ
assert (x ∈ @singleton _ EVarSet _ y <-> x = y) by set_solver;
tauto.
Qed.

Corollary free_evars_evar_open ϕ x dbi :
free_evars (evar_open dbi x ϕ) ⊆ union (singleton x) (free_evars ϕ).
Σ: Signature
ϕ: Pattern
x: evar
dbi: db_index
free_evars (evar_open dbi x ϕ) ⊆ {[x]} ∪ free_evars ϕ
Proof.
Σ: Signature
ϕ: Pattern
x: evar
dbi: db_index
free_evars (evar_open dbi x ϕ) ⊆ {[x]} ∪ free_evars ϕ
apply free_evars_bevar_subst; auto.
Qed.

Lemma free_evars_evar_open' ϕ x dbi:
free_evars ϕ ⊆ free_evars (evar_open dbi x ϕ).
Σ: Signature
ϕ: Pattern
x: evar
dbi: db_index
free_evars ϕ ⊆ free_evars (evar_open dbi x ϕ)
Proof.
Σ: Signature
ϕ: Pattern
x: evar
dbi: db_index
free_evars ϕ ⊆ free_evars (evar_open dbi x ϕ)
move: dbi.
Σ: Signature
ϕ: Pattern
x: evar
∀ dbi : db_index,
  free_evars ϕ ⊆ free_evars (evar_open dbi x ϕ)
induction ϕ; intros dbi; simpl; try apply empty_subseteq.
Σ: Signature
x0, x: evar
dbi: db_index
{[x0]} ⊆ {[x0]}
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ1 ∪ free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ1 ∪ free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
Σ: Signature
ϕ: Pattern
x: evar
IHϕ: ∀ dbi : db_index,
  free_evars ϕ ⊆ free_evars (evar_open dbi x ϕ)
dbi: db_index
free_evars ϕ
⊆ free_evars ϕ^[evar:S dbi↦patt_free_evar x]
Σ: Signature
ϕ: Pattern
x: evar
IHϕ: ∀ dbi : db_index,
  free_evars ϕ ⊆ free_evars (evar_open dbi x ϕ)
dbi: db_index
free_evars ϕ
⊆ free_evars ϕ^[evar:dbi↦patt_free_evar x]
-
Σ: Signature
x0, x: evar
dbi: db_index
{[x0]} ⊆ {[x0]} apply PreOrder_Reflexive.
-
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ1 ∪ free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] apply union_least.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ1
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
  +
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ1
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] eapply PreOrder_Transitive.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ1 ⊆ ?y
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
?y
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] apply IHϕ1.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars (evar_open ?dbi x ϕ1)
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
    apply union_subseteq_l.
  +
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] eapply PreOrder_Transitive.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ2 ⊆ ?y
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
?y
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] apply IHϕ2.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars (evar_open ?dbi x ϕ2)
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
    apply union_subseteq_r.
-
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ1 ∪ free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] apply union_least.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ1
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
  +
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ1
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] eapply PreOrder_Transitive.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ1 ⊆ ?y
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
?y
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] apply IHϕ1.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars (evar_open ?dbi x ϕ1)
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
    apply union_subseteq_l.
  +
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ2
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] eapply PreOrder_Transitive.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars ϕ2 ⊆ ?y
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
?y
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x] apply IHϕ2.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ dbi : db_index,
  free_evars ϕ1
  ⊆ free_evars (evar_open dbi x ϕ1)
IHϕ2: ∀ dbi : db_index,
  free_evars ϕ2
  ⊆ free_evars (evar_open dbi x ϕ2)
dbi: db_index
free_evars (evar_open ?dbi x ϕ2)
⊆ free_evars ϕ1^[evar:dbi↦patt_free_evar x]
  ∪ free_evars ϕ2^[evar:dbi↦patt_free_evar x]
    apply union_subseteq_r.
-
Σ: Signature
ϕ: Pattern
x: evar
IHϕ: ∀ dbi : db_index, free_evars ϕ ⊆ free_evars (evar_open dbi x ϕ)
dbi: db_index
free_evars ϕ
⊆ free_evars ϕ^[evar:S dbi↦patt_free_evar x] set_solver.
-
Σ: Signature
ϕ: Pattern
x: evar
IHϕ: ∀ dbi : db_index, free_evars ϕ ⊆ free_evars (evar_open dbi x ϕ)
dbi: db_index
free_evars ϕ
⊆ free_evars ϕ^[evar:dbi↦patt_free_evar x] set_solver.
Qed.

Lemma free_evar_subst_no_occurrence x p q:
count_evar_occurrences x p = 0 ->
free_evar_subst p q x = p.
Σ: Signature
x: evar
p, q: Pattern
count_evar_occurrences x p = 0 → p^[[evar:x↦q]] = p
Proof.
Σ: Signature
x: evar
p, q: Pattern
count_evar_occurrences x p = 0 → p^[[evar:x↦q]] = p
intros H.
Σ: Signature
x: evar
p, q: Pattern
H: count_evar_occurrences x p = 0
p^[[evar:x↦q]] = p
remember (size' p) as sz.
Σ: Signature
x: evar
p, q: Pattern
H: count_evar_occurrences x p = 0
sz: nat
Heqsz: sz = size' p
p^[[evar:x↦q]] = p
assert (Hsz: size' p <= sz) by lia.
Σ: Signature
x: evar
p, q: Pattern
H: count_evar_occurrences x p = 0
sz: nat
Heqsz: sz = size' p
Hsz: size' p ≤ sz
p^[[evar:x↦q]] = p
clear Heqsz.
Σ: Signature
x: evar
p, q: Pattern
H: count_evar_occurrences x p = 0
sz: nat
Hsz: size' p ≤ sz
p^[[evar:x↦q]] = p

move: p Hsz H.
Σ: Signature
x: evar
q: Pattern
sz: nat
∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
induction sz; intros p Hsz H; destruct p; simpl in *; try lia; auto.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: (if decide (x0 = x) then 1 else 0) = 0
(if decide (x = x0) then q else patt_free_evar x0) =
patt_free_evar x0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_app p1^[[evar:x↦q]] p2^[[evar:x↦q]] =
patt_app p1 p2
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_imp p1^[[evar:x↦q]] p2^[[evar:x↦q]] =
patt_imp p1 p2
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: count_evar_occurrences x p = 0
patt_exists p^[[evar:x↦q]] = patt_exists p
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: count_evar_occurrences x p = 0
patt_mu p^[[evar:x↦q]] = patt_mu p
-
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: (if decide (x0 = x) then 1 else 0) = 0
(if decide (x = x0) then q else patt_free_evar x0) =
patt_free_evar x0 simpl in H.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: (if decide (x0 = x) then 1 else 0) = 0
(if decide (x = x0) then q else patt_free_evar x0) =
patt_free_evar x0 simpl.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: (if decide (x0 = x) then 1 else 0) = 0
(if decide (x = x0) then q else patt_free_evar x0) =
patt_free_evar x0
  destruct (decide (x = x0)).
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: (if decide (x0 = x) then 1 else 0) = 0
e: x = x0
q = patt_free_evar x0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: (if decide (x0 = x) then 1 else 0) = 0
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0
  +
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: (if decide (x0 = x) then 1 else 0) = 0
e: x = x0
q = patt_free_evar x0 subst x0.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
Hsz: 1 ≤ S sz
H: (if decide (x = x) then 1 else 0) = 0
q = patt_free_evar x destruct (decide (x = x)).
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
Hsz: 1 ≤ S sz
e: x = x
H: 1 = 0
q = patt_free_evar x
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
Hsz: 1 ≤ S sz
n: x ≠ x
H: 0 = 0
q = patt_free_evar x simpl in H.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
Hsz: 1 ≤ S sz
e: x = x
H: 1 = 0
q = patt_free_evar x
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
Hsz: 1 ≤ S sz
n: x ≠ x
H: 0 = 0
q = patt_free_evar x inversion H.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
Hsz: 1 ≤ S sz
n: x ≠ x
H: 0 = 0
q = patt_free_evar x contradiction.
  +
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
x0: evar
Hsz: 1 ≤ S sz
H: (if decide (x0 = x) then 1 else 0) = 0
n: x ≠ x0
patt_free_evar x0 = patt_free_evar x0 reflexivity.
-
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_app p1^[[evar:x↦q]] p2^[[evar:x↦q]] =
patt_app p1 p2 rewrite IHsz.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
size' p1 ≤ sz
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
count_evar_occurrences x p1 = 0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_app p1 p2^[[evar:x↦q]] = patt_app p1 p2 lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
count_evar_occurrences x p1 = 0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_app p1 p2^[[evar:x↦q]] = patt_app p1 p2 lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_app p1 p2^[[evar:x↦q]] = patt_app p1 p2 rewrite IHsz.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
size' p2 ≤ sz
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
count_evar_occurrences x p2 = 0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_app p1 p2 = patt_app p1 p2 lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
count_evar_occurrences x p2 = 0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_app p1 p2 = patt_app p1 p2 lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_app p1 p2 = patt_app p1 p2 reflexivity.
-
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_imp p1^[[evar:x↦q]] p2^[[evar:x↦q]] =
patt_imp p1 p2 rewrite IHsz.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
size' p1 ≤ sz
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
count_evar_occurrences x p1 = 0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_imp p1 p2^[[evar:x↦q]] = patt_imp p1 p2 lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
count_evar_occurrences x p1 = 0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_imp p1 p2^[[evar:x↦q]] = patt_imp p1 p2 lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_imp p1 p2^[[evar:x↦q]] = patt_imp p1 p2 rewrite IHsz.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
size' p2 ≤ sz
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
count_evar_occurrences x p2 = 0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_imp p1 p2 = patt_imp p1 p2 lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
count_evar_occurrences x p2 = 0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_imp p1 p2 = patt_imp p1 p2 lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p1, p2: Pattern
Hsz: S (size' p1 + size' p2) ≤ S sz
H: count_evar_occurrences x p1 +
count_evar_occurrences x p2 = 0
patt_imp p1 p2 = patt_imp p1 p2 reflexivity.
-
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: count_evar_occurrences x p = 0
patt_exists p^[[evar:x↦q]] = patt_exists p f_equal.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: count_evar_occurrences x p = 0
p^[[evar:x↦q]] = p rewrite IHsz.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: count_evar_occurrences x p = 0
size' p ≤ sz
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: count_evar_occurrences x p = 0
count_evar_occurrences x p = 0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: count_evar_occurrences x p = 0
p = p lia.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: count_evar_occurrences x p = 0
count_evar_occurrences x p = 0
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: count_evar_occurrences x p = 0
p = p exact H.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: count_evar_occurrences x p = 0
p = p reflexivity.
-
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: count_evar_occurrences x p = 0
patt_mu p^[[evar:x↦q]] = patt_mu p rewrite IHsz; auto.
Σ: Signature
x: evar
q: Pattern
sz: nat
IHsz: ∀ p : Pattern,
  size' p ≤ sz
  → count_evar_occurrences x p = 0
    → p^[[evar:x↦q]] = p
p: Pattern
Hsz: S (size' p) ≤ S sz
H: count_evar_occurrences x p = 0
size' p ≤ sz lia.
Qed.


Lemma Private_bsvar_occur_evar_open sz dbi1 dbi2 X phi:
size phi <= sz ->
bsvar_occur phi dbi1 = false ->
bsvar_occur (evar_open dbi2 X phi) dbi1 = false.
Σ: Signature
sz: nat
dbi1, dbi2: db_index
X: evar
phi: Pattern
size phi ≤ sz
→ bsvar_occur phi dbi1 = false
  → bsvar_occur (evar_open dbi2 X phi) dbi1 = false
Proof.
Σ: Signature
sz: nat
dbi1, dbi2: db_index
X: evar
phi: Pattern
size phi ≤ sz
→ bsvar_occur phi dbi1 = false
  → bsvar_occur (evar_open dbi2 X phi) dbi1 = false
move: phi dbi1 dbi2.
Σ: Signature
sz: nat
X: evar
∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bsvar_occur phi dbi1 = false
    → bsvar_occur (evar_open dbi2 X phi) dbi1 = false
induction sz; move=> phi; destruct phi; simpl; move=> dbi1 dbi2 Hsz H; try rewrite !IHsz; auto; try lia; try apply orb_false_elim in H; firstorder.
Σ: Signature
X: evar
n, dbi1, dbi2: db_index
Hsz: 0 ≤ 0
H: false = false
bsvar_occur (evar_open dbi2 X (patt_bound_evar n))
  dbi1 = false
Σ: Signature
sz: nat
X: evar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bsvar_occur phi dbi1 = false
    → bsvar_occur (evar_open dbi2 X phi) dbi1 =
      false
n, dbi1, dbi2: db_index
Hsz: 0 ≤ S sz
H: false = false
size (patt_bound_evar n) ≤ sz
2: {
Σ: Signature
sz: nat
X: evar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bsvar_occur phi dbi1 = false
    → bsvar_occur (evar_open dbi2 X phi) dbi1 =
      false
n, dbi1, dbi2: db_index
Hsz: 0 ≤ S sz
H: false = false
size (patt_bound_evar n) ≤ sz simpl.
Σ: Signature
sz: nat
X: evar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bsvar_occur phi dbi1 = false
    → bsvar_occur (evar_open dbi2 X phi) dbi1 =
      false
n, dbi1, dbi2: db_index
Hsz: 0 ≤ S sz
H: false = false
0 ≤ sz lia. }
Σ: Signature
X: evar
n, dbi1, dbi2: db_index
Hsz: 0 ≤ 0
H: false = false
bsvar_occur (evar_open dbi2 X (patt_bound_evar n))
  dbi1 = false
cbn.
Σ: Signature
X: evar
n, dbi1, dbi2: db_index
Hsz: 0 ≤ 0
H: false = false
bsvar_occur
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar X
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end dbi1 = false case_match; reflexivity.
Qed.

Corollary bsvar_occur_evar_open dbi1 dbi2 X phi:
bsvar_occur phi dbi1 = false ->
bsvar_occur (evar_open dbi2 X phi) dbi1 = false.
Σ: Signature
dbi1, dbi2: db_index
X: evar
phi: Pattern
bsvar_occur phi dbi1 = false
→ bsvar_occur (evar_open dbi2 X phi) dbi1 = false
Proof.
Σ: Signature
dbi1, dbi2: db_index
X: evar
phi: Pattern
bsvar_occur phi dbi1 = false
→ bsvar_occur (evar_open dbi2 X phi) dbi1 = false
apply Private_bsvar_occur_evar_open with (sz := size phi).
Σ: Signature
dbi1, dbi2: db_index
X: evar
phi: Pattern
size phi ≤ size phi lia.
Qed.

Lemma Private_bevar_occur_svar_open sz dbi1 dbi2 X phi:
size phi <= sz ->
bevar_occur phi dbi1 = false ->
bevar_occur (svar_open dbi2 X phi) dbi1 = false.
Σ: Signature
sz: nat
dbi1, dbi2: db_index
X: svar
phi: Pattern
size phi ≤ sz
→ bevar_occur phi dbi1 = false
  → bevar_occur (svar_open dbi2 X phi) dbi1 = false
Proof.
Σ: Signature
sz: nat
dbi1, dbi2: db_index
X: svar
phi: Pattern
size phi ≤ sz
→ bevar_occur phi dbi1 = false
  → bevar_occur (svar_open dbi2 X phi) dbi1 = false
move: phi dbi1 dbi2.
Σ: Signature
sz: nat
X: svar
∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bevar_occur phi dbi1 = false
    → bevar_occur (svar_open dbi2 X phi) dbi1 = false
induction sz; move=> phi; destruct phi; simpl; move=> dbi1 dbi2 Hsz H; try rewrite !IHsz; auto; try lia; try apply orb_false_elim in H; firstorder.
Σ: Signature
X: svar
n, dbi1, dbi2: db_index
Hsz: 0 ≤ 0
H: false = false
bevar_occur (svar_open dbi2 X (patt_bound_svar n))
  dbi1 = false
Σ: Signature
sz: nat
X: svar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bevar_occur phi dbi1 = false
    → bevar_occur (svar_open dbi2 X phi) dbi1 =
      false
n, dbi1, dbi2: db_index
Hsz: 0 ≤ S sz
H: false = false
size (patt_bound_svar n) ≤ sz
{
Σ: Signature
X: svar
n, dbi1, dbi2: db_index
Hsz: 0 ≤ 0
H: false = false
bevar_occur (svar_open dbi2 X (patt_bound_svar n))
  dbi1 = false cbn.
Σ: Signature
X: svar
n, dbi1, dbi2: db_index
Hsz: 0 ≤ 0
H: false = false
bevar_occur
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => patt_free_svar X
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end dbi1 = false case_match; reflexivity. }
Σ: Signature
sz: nat
X: svar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bevar_occur phi dbi1 = false
    → bevar_occur (svar_open dbi2 X phi) dbi1 =
      false
n, dbi1, dbi2: db_index
Hsz: 0 ≤ S sz
H: false = false
size (patt_bound_svar n) ≤ sz
simpl.
Σ: Signature
sz: nat
X: svar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : db_index),
  size phi ≤ sz
  → bevar_occur phi dbi1 = false
    → bevar_occur (svar_open dbi2 X phi) dbi1 =
      false
n, dbi1, dbi2: db_index
Hsz: 0 ≤ S sz
H: false = false
0 ≤ sz lia.
Qed.

Corollary bevar_occur_svar_open dbi1 dbi2 X phi:
bevar_occur phi dbi1 = false ->
bevar_occur (svar_open dbi2 X phi) dbi1 = false.
Σ: Signature
dbi1, dbi2: db_index
X: svar
phi: Pattern
bevar_occur phi dbi1 = false
→ bevar_occur (svar_open dbi2 X phi) dbi1 = false
Proof.
Σ: Signature
dbi1, dbi2: db_index
X: svar
phi: Pattern
bevar_occur phi dbi1 = false
→ bevar_occur (svar_open dbi2 X phi) dbi1 = false
apply Private_bevar_occur_svar_open with (sz := size phi).
Σ: Signature
dbi1, dbi2: db_index
X: svar
phi: Pattern
size phi ≤ size phi lia.
Qed.

Lemma Private_bevar_occur_evar_open sz dbi1 dbi2 X phi:
size phi <= sz -> dbi1 < dbi2 ->
bevar_occur phi dbi1 = false ->
bevar_occur (evar_open dbi2 X phi) dbi1 = false.
Σ: Signature
sz, dbi1, dbi2: nat
X: evar
phi: Pattern
size phi ≤ sz
→ dbi1 < dbi2
  → bevar_occur phi dbi1 = false
    → bevar_occur (evar_open dbi2 X phi) dbi1 = false
Proof.
Σ: Signature
sz, dbi1, dbi2: nat
X: evar
phi: Pattern
size phi ≤ sz
→ dbi1 < dbi2
  → bevar_occur phi dbi1 = false
    → bevar_occur (evar_open dbi2 X phi) dbi1 = false
move: phi dbi1 dbi2.
Σ: Signature
sz: nat
X: evar
∀ (phi : Pattern) (dbi1 dbi2 : nat),
  size phi ≤ sz
  → dbi1 < dbi2
    → bevar_occur phi dbi1 = false
      → bevar_occur (evar_open dbi2 X phi) dbi1 =
        false
induction sz; move=> phi; destruct phi; simpl; move=> dbi1 dbi2 Hsz H H1; try rewrite !IHsz; auto; try lia; try apply orb_false_elim in H1; firstorder.
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
H1: (if decide (n = dbi1) then true else false) =
false
bevar_occur (evar_open dbi2 X (patt_bound_evar n))
  dbi1 = false
Σ: Signature
sz: nat
X: evar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : nat),
  size phi ≤ sz
  → dbi1 < dbi2
    → bevar_occur phi dbi1 = false
      → bevar_occur (evar_open dbi2 X phi) dbi1 =
        false
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ S sz
H: dbi1 < dbi2
H1: (if decide (n = dbi1) then true else false) =
false
size (patt_bound_evar n) ≤ sz
{
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
H1: (if decide (n = dbi1) then true else false) =
false
bevar_occur (evar_open dbi2 X (patt_bound_evar n))
  dbi1 = false cbn.
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
H1: (if decide (n = dbi1) then true else false) =
false
bevar_occur
  match compare_nat n dbi2 with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar X
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end dbi1 = false repeat case_match; simpl; auto; try lia.
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H1: false = false
l: n < dbi2
H2: compare_nat n dbi2 = Nat_less n dbi2 l
(if decide (n = dbi1) then true else false) = false
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H1: false = false
g: n > dbi2
H2: compare_nat n dbi2 = Nat_greater n dbi2 g
(if decide (Nat.pred n = dbi1) then true else false) =
false
  {
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H1: false = false
l: n < dbi2
H2: compare_nat n dbi2 = Nat_less n dbi2 l
(if decide (n = dbi1) then true else false) = false rewrite H0.
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H1: false = false
l: n < dbi2
H2: compare_nat n dbi2 = Nat_less n dbi2 l
false = false reflexivity. }
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H1: false = false
g: n > dbi2
H2: compare_nat n dbi2 = Nat_greater n dbi2 g
(if decide (Nat.pred n = dbi1) then true else false) =
false
  {
Σ: Signature
X: evar
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ 0
H: dbi1 < dbi2
n0: n ≠ dbi1
H0: decide (n = dbi1) = right n0
H1: false = false
g: n > dbi2
H2: compare_nat n dbi2 = Nat_greater n dbi2 g
(if decide (Nat.pred n = dbi1) then true else false) =
false case_match; try lia. }
}
Σ: Signature
sz: nat
X: evar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : nat),
  size phi ≤ sz
  → dbi1 < dbi2
    → bevar_occur phi dbi1 = false
      → bevar_occur (evar_open dbi2 X phi) dbi1 =
        false
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ S sz
H: dbi1 < dbi2
H1: (if decide (n = dbi1) then true else false) =
false
size (patt_bound_evar n) ≤ sz
simpl.
Σ: Signature
sz: nat
X: evar
IHsz: ∀ (phi : Pattern) (dbi1 dbi2 : nat),
  size phi ≤ sz
  → dbi1 < dbi2
    → bevar_occur phi dbi1 = false
      → bevar_occur (evar_open dbi2 X phi) dbi1 =
        false
n: db_index
dbi1, dbi2: nat
Hsz: 0 ≤ S sz
H: dbi1 < dbi2
H1: (if decide (n = dbi1) then true else false) =
false
0 ≤ sz lia.
Qed.

Corollary bevar_occur_evar_open dbi1 dbi2 X phi:
bevar_occur phi dbi1 = false -> dbi1 < dbi2 ->
bevar_occur (evar_open dbi2 X phi) dbi1 = false.
Σ: Signature
dbi1: db_index
dbi2: nat
X: evar
phi: Pattern
bevar_occur phi dbi1 = false
→ dbi1 < dbi2
  → bevar_occur (evar_open dbi2 X phi) dbi1 = false
Proof.
Σ: Signature
dbi1: db_index
dbi2: nat
X: evar
phi: Pattern
bevar_occur phi dbi1 = false
→ dbi1 < dbi2
  → bevar_occur (evar_open dbi2 X phi) dbi1 = false
intros H H0.
Σ: Signature
dbi1: db_index
dbi2: nat
X: evar
phi: Pattern
H: bevar_occur phi dbi1 = false
H0: dbi1 < dbi2
bevar_occur (evar_open dbi2 X phi) dbi1 = false apply Private_bevar_occur_evar_open with (sz := size phi); auto.
Qed.

Lemma well_formed_positive_bevar_subst φ : forall n ψ,
mu_free φ ->
well_formed_positive φ = true -> well_formed_positive ψ = true
->
well_formed_positive (bevar_subst φ ψ n) = true.
Σ: Signature
φ: Pattern
∀ (n : db_index) (ψ : Pattern),
  mu_free φ
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
Proof.
Σ: Signature
φ: Pattern
∀ (n : db_index) (ψ : Pattern),
  mu_free φ
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
induction φ; intros n' ψ H H0 H1; simpl; auto.
Σ: Signature
n, n': db_index
ψ: Pattern
H: mu_free (patt_bound_evar n)
H0: well_formed_positive (patt_bound_evar n) = true
H1: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (ψ : Pattern),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (n : db_index) (ψ : Pattern),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
n': db_index
ψ: Pattern
H: mu_free (patt_app φ1 φ2)
H0: well_formed_positive (patt_app φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (ψ : Pattern),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (n : db_index) (ψ : Pattern),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
n': db_index
ψ: Pattern
H: mu_free (patt_imp φ1 φ2)
H0: well_formed_positive (patt_imp φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
2-3: apply andb_true_iff in H as [E1 E2];
     simpl in H0; apply eq_sym, andb_true_eq in H0; destruct H0; 
     rewrite -> IHφ1, -> IHφ2; auto.
Σ: Signature
n, n': db_index
ψ: Pattern
H: mu_free (patt_bound_evar n)
H0: well_formed_positive (patt_bound_evar n) = true
H1: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true
*
Σ: Signature
n, n': db_index
ψ: Pattern
H: mu_free (patt_bound_evar n)
H0: well_formed_positive (patt_bound_evar n) = true
H1: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true break_match_goal; auto.
Qed.

Lemma mu_free_bevar_subst :
forall φ ψ, mu_free φ -> mu_free ψ -> forall n, mu_free (bevar_subst φ ψ n).
Σ: Signature
∀ φ ψ : Pattern,
  mu_free φ
  → mu_free ψ → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
Proof.
Σ: Signature
∀ φ ψ : Pattern,
  mu_free φ
  → mu_free ψ → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
induction φ; intros ψ H H0 n'; simpl; try now constructor.
Σ: Signature
n: db_index
ψ: Pattern
H: mu_free (patt_bound_evar n)
H0: mu_free ψ
n': db_index
mu_free
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_app φ1 φ2)
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_imp φ1 φ2)
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ]
Σ: Signature
φ: Pattern
IHφ: ∀ ψ : Pattern,
  mu_free φ
  → mu_free ψ
    → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_exists φ)
H0: mu_free ψ
n': db_index
mu_free φ^[evar:S n'↦ψ]
Σ: Signature
φ: Pattern
IHφ: ∀ ψ : Pattern,
  mu_free φ
  → mu_free ψ
    → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_mu φ)
H0: mu_free ψ
n': db_index
false
*
Σ: Signature
n: db_index
ψ: Pattern
H: mu_free (patt_bound_evar n)
H0: mu_free ψ
n': db_index
mu_free
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end break_match_goal; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_app φ1 φ2)
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ] simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
H: mu_free φ1 && mu_free φ2
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ] apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
E1: mu_free φ1 = true
E2: mu_free φ2 = true
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ] now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_imp φ1 φ2)
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ] simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
H: mu_free φ1 && mu_free φ2
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ] apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ ψ : Pattern,
  mu_free φ1
  → mu_free ψ
    → ∀ n : db_index, mu_free φ1^[evar:n↦ψ]
IHφ2: ∀ ψ : Pattern,
  mu_free φ2
  → mu_free ψ
    → ∀ n : db_index, mu_free φ2^[evar:n↦ψ]
ψ: Pattern
E1: mu_free φ1 = true
E2: mu_free φ2 = true
H0: mu_free ψ
n': db_index
mu_free φ1^[evar:n'↦ψ] && mu_free φ2^[evar:n'↦ψ] now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ ψ : Pattern, mu_free φ → mu_free ψ → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_exists φ)
H0: mu_free ψ
n': db_index
mu_free φ^[evar:S n'↦ψ] simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ ψ : Pattern, mu_free φ → mu_free ψ → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
ψ: Pattern
H: mu_free φ
H0: mu_free ψ
n': db_index
mu_free φ^[evar:S n'↦ψ] now apply IHφ.
*
Σ: Signature
φ: Pattern
IHφ: ∀ ψ : Pattern, mu_free φ → mu_free ψ → ∀ n : db_index, mu_free φ^[evar:n↦ψ]
ψ: Pattern
H: mu_free (patt_mu φ)
H0: mu_free ψ
n': db_index
false inversion H.
Qed.

Corollary mu_free_evar_open :
forall φ, mu_free φ -> forall x n, mu_free (evar_open n x φ).
Σ: Signature
∀ φ : Pattern,
  mu_free φ
  → ∀ (x : evar) (n : db_index),
      mu_free (evar_open n x φ)
Proof.
Σ: Signature
∀ φ : Pattern,
  mu_free φ
  → ∀ (x : evar) (n : db_index),
      mu_free (evar_open n x φ)
intros φ H x n.
Σ: Signature
φ: Pattern
H: mu_free φ
x: evar
n: db_index
mu_free (evar_open n x φ) apply mu_free_bevar_subst; auto.
Qed.

Theorem evar_open_free_evar_subst_swap :
forall φ x n ψ y, x <> y -> well_formed ψ ->
  evar_open n x (free_evar_subst φ ψ y) = free_evar_subst (evar_open n x φ) ψ y.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index) (ψ : Pattern) 
  (y : evar),
  x ≠ y
  → well_formed ψ
    → evar_open n x φ^[[evar:y↦ψ]] =
      (evar_open n x φ)^[[evar:y↦ψ]]
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index) (ψ : Pattern) 
  (y : evar),
  x ≠ y
  → well_formed ψ
    → evar_open n x φ^[[evar:y↦ψ]] =
      (evar_open n x φ)^[[evar:y↦ψ]]
induction φ; intros x' n' ψ y H H0; simpl; auto.
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
evar_open n' x'
  (if decide (y = x) then ψ else patt_free_evar x) =
(if decide (y = x) then ψ else patt_free_evar x)
Σ: Signature
n: db_index
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
evar_open n' x' (patt_bound_evar n) =
(evar_open n' x' (patt_bound_evar n))^[[evar:y↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → evar_open n x φ1^[[evar:y↦ψ]] =
      (evar_open n x φ1)^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → evar_open n x φ2^[[evar:y↦ψ]] =
      (evar_open n x φ2)^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
evar_open n' x'
  (patt_app φ1^[[evar:y↦ψ]] φ2^[[evar:y↦ψ]]) =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → evar_open n x φ1^[[evar:y↦ψ]] =
      (evar_open n x φ1)^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → evar_open n x φ2^[[evar:y↦ψ]] =
      (evar_open n x φ2)^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
evar_open n' x'
  (patt_imp φ1^[[evar:y↦ψ]] φ2^[[evar:y↦ψ]]) =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → evar_open n x φ^[[evar:y↦ψ]] =
      (evar_open n x φ)^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
evar_open n' x' (patt_exists φ^[[evar:y↦ψ]]) =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:y↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → evar_open n x φ^[[evar:y↦ψ]] =
      (evar_open n x φ)^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
evar_open n' x' (patt_mu φ^[[evar:y↦ψ]]) =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
*
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
evar_open n' x'
  (if decide (y = x) then ψ else patt_free_evar x) =
(if decide (y = x) then ψ else patt_free_evar x) destruct (decide (y = x)); simpl.
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
e: y = x
evar_open n' x' ψ = ψ
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
n: y ≠ x
evar_open n' x' (patt_free_evar x) = patt_free_evar x
  **
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
e: y = x
evar_open n' x' ψ = ψ rewrite evar_open_wfc; auto.
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
e: y = x
well_formed_closed_ex_aux ψ 0 unfold well_formed,well_formed_closed in H0.
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: [&& well_formed_positive ψ,
    well_formed_closed_mu_aux ψ 0
  & well_formed_closed_ex_aux ψ 0]
e: y = x
well_formed_closed_ex_aux ψ 0 destruct_and!.
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H1: well_formed_positive ψ = true
H0: well_formed_closed_mu_aux ψ 0 = true
H3: well_formed_closed_ex_aux ψ 0 = true
e: y = x
well_formed_closed_ex_aux ψ 0
     assumption.
  **
Σ: Signature
x, x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
n: y ≠ x
evar_open n' x' (patt_free_evar x) = patt_free_evar x reflexivity.
*
Σ: Signature
n: db_index
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
evar_open n' x' (patt_bound_evar n) =
(evar_open n' x' (patt_bound_evar n))^[[evar:y↦ψ]] cbn.
Σ: Signature
n: db_index
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x'
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x'
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[[evar:y↦ψ]] break_match_goal; simpl; auto.
Σ: Signature
n: db_index
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
e: n = n'
Heqc: compare_nat n n' = Nat_equal n n' e
patt_free_evar x' =
(if decide (y = x') then ψ else patt_free_evar x') destruct (decide (y = x')); auto.
Σ: Signature
n: db_index
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
e: n = n'
Heqc: compare_nat n n' = Nat_equal n n' e
e0: y = x'
patt_free_evar x' = ψ
  congruence.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → evar_open n x φ1^[[evar:y↦ψ]] =
      (evar_open n x φ1)^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → evar_open n x φ2^[[evar:y↦ψ]] =
      (evar_open n x φ2)^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
evar_open n' x'
  (patt_app φ1^[[evar:y↦ψ]] φ2^[[evar:y↦ψ]]) =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] unfold evar_open in *.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ1^[[evar:y↦ψ]]^[evar:n↦
      patt_free_evar x] =
      φ1^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ2^[[evar:y↦ψ]]^[evar:n↦
      patt_free_evar x] =
      φ2^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_app φ1^[[evar:y↦ψ]] φ2^[[evar:y↦ψ]])^[evar:n'↦
patt_free_evar x'] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] now rewrite -> bevar_subst_app, -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → evar_open n x φ1^[[evar:y↦ψ]] =
      (evar_open n x φ1)^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → evar_open n x φ2^[[evar:y↦ψ]] =
      (evar_open n x φ2)^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
evar_open n' x'
  (patt_imp φ1^[[evar:y↦ψ]] φ2^[[evar:y↦ψ]]) =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] unfold evar_open in *.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ1^[[evar:y↦ψ]]^[evar:n↦
      patt_free_evar x] =
      φ1^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
IHφ2: ∀ (x : evar) (n : db_index) 
  (ψ : Pattern) (y : evar),
  x ≠ y
  → well_formed ψ
    → φ2^[[evar:y↦ψ]]^[evar:n↦
      patt_free_evar x] =
      φ2^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_imp φ1^[[evar:y↦ψ]] φ2^[[evar:y↦ψ]])^[evar:n'↦
patt_free_evar x'] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] now rewrite -> bevar_subst_imp, -> IHφ1, -> IHφ2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) (ψ : Pattern) (y : evar),
x ≠ y
→ well_formed ψ
→ evar_open n x φ^[[evar:y↦ψ]] = (evar_open n x φ)^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
evar_open n' x' (patt_exists φ^[[evar:y↦ψ]]) =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:y↦ψ]] unfold evar_open in *.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) (ψ : Pattern) (y : evar),
x ≠ y
→ well_formed ψ
→ φ^[[evar:y↦ψ]]^[evar:n↦patt_free_evar x] =
φ^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_exists φ^[[evar:y↦ψ]])^[evar:n'↦patt_free_evar
                                        x'] =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:y↦ψ]] now rewrite -> bevar_subst_exists, -> IHφ.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) (ψ : Pattern) (y : evar),
x ≠ y
→ well_formed ψ
→ evar_open n x φ^[[evar:y↦ψ]] = (evar_open n x φ)^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
evar_open n' x' (patt_mu φ^[[evar:y↦ψ]]) =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] unfold evar_open in *.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index) (ψ : Pattern) (y : evar),
x ≠ y
→ well_formed ψ
→ φ^[[evar:y↦ψ]]^[evar:n↦patt_free_evar x] =
φ^[evar:n↦patt_free_evar x]^[[evar:y↦ψ]]
x': evar
n': db_index
ψ: Pattern
y: evar
H: x' ≠ y
H0: well_formed ψ
(patt_mu φ^[[evar:y↦ψ]])^[evar:n'↦patt_free_evar x'] =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:y↦ψ]] now rewrite -> bevar_subst_mu, -> IHφ.
Qed.

Lemma free_evars_free_evar_subst : forall φ ψ x,
free_evars (free_evar_subst φ ψ x) ⊆ free_evars φ ∪ free_evars ψ.
Σ: Signature
∀ (φ ψ : Pattern) (x : evar),
  free_evars φ^[[evar:x↦ψ]]
  ⊆ free_evars φ ∪ free_evars ψ
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (x : evar),
  free_evars φ^[[evar:x↦ψ]]
  ⊆ free_evars φ ∪ free_evars ψ
induction φ; intros ψ x'; simpl.
Σ: Signature
x: evar
ψ: Pattern
x': evar
free_evars
  (if decide (x' = x) then ψ else patt_free_evar x)
⊆ {[x]} ∪ free_evars ψ
Σ: Signature
x: svar
ψ: Pattern
x': evar
∅ ⊆ ∅ ∪ free_evars ψ
Σ: Signature
n: db_index
ψ: Pattern
x': evar
∅ ⊆ ∅ ∪ free_evars ψ
Σ: Signature
n: db_index
ψ: Pattern
x': evar
∅ ⊆ ∅ ∪ free_evars ψ
Σ: Signature
sigma: symbols
ψ: Pattern
x': evar
∅ ⊆ ∅ ∪ free_evars ψ
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (x : evar),
  free_evars φ1^[[evar:x↦ψ]]
  ⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ
Σ: Signature
ψ: Pattern
x': evar
∅ ⊆ ∅ ∪ free_evars ψ
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (x : evar),
  free_evars φ1^[[evar:x↦ψ]]
  ⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (x : evar),
  free_evars φ^[[evar:x↦ψ]]
  ⊆ free_evars φ ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ^[[evar:x'↦ψ]]
⊆ free_evars φ ∪ free_evars ψ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (x : evar),
  free_evars φ^[[evar:x↦ψ]]
  ⊆ free_evars φ ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ^[[evar:x'↦ψ]]
⊆ free_evars φ ∪ free_evars ψ
2-5, 7: apply empty_subseteq.
Σ: Signature
x: evar
ψ: Pattern
x': evar
free_evars
  (if decide (x' = x) then ψ else patt_free_evar x)
⊆ {[x]} ∪ free_evars ψ
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (x : evar),
  free_evars φ1^[[evar:x↦ψ]]
  ⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (x : evar),
  free_evars φ1^[[evar:x↦ψ]]
  ⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (x : evar),
  free_evars φ^[[evar:x↦ψ]]
  ⊆ free_evars φ ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ^[[evar:x'↦ψ]]
⊆ free_evars φ ∪ free_evars ψ
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (x : evar),
  free_evars φ^[[evar:x↦ψ]]
  ⊆ free_evars φ ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ^[[evar:x'↦ψ]]
⊆ free_evars φ ∪ free_evars ψ
*
Σ: Signature
x: evar
ψ: Pattern
x': evar
free_evars
  (if decide (x' = x) then ψ else patt_free_evar x)
⊆ {[x]} ∪ free_evars ψ destruct (decide (x' = x)); simpl.
Σ: Signature
x: evar
ψ: Pattern
x': evar
e: x' = x
free_evars ψ ⊆ {[x]} ∪ free_evars ψ
Σ: Signature
x: evar
ψ: Pattern
x': evar
n: x' ≠ x
{[x]} ⊆ {[x]} ∪ free_evars ψ
  **
Σ: Signature
x: evar
ψ: Pattern
x': evar
e: x' = x
free_evars ψ ⊆ {[x]} ∪ free_evars ψ apply union_subseteq_r.
  **
Σ: Signature
x: evar
ψ: Pattern
x': evar
n: x' ≠ x
{[x]} ⊆ {[x]} ∪ free_evars ψ apply union_subseteq_l.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (x : evar),
  free_evars φ1^[[evar:x↦ψ]]
  ⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ specialize (IHφ1 ψ x').
Σ: Signature
φ1, φ2, ψ: Pattern
x': evar
IHφ1: free_evars φ1^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ specialize (IHφ2 ψ x').
Σ: Signature
φ1, φ2, ψ: Pattern
x': evar
IHφ1: free_evars φ1^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ2 ∪ free_evars ψ
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ
  set_solver.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (x : evar),
  free_evars φ1^[[evar:x↦ψ]]
  ⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ specialize (IHφ1 ψ x').
Σ: Signature
φ1, φ2, ψ: Pattern
x': evar
IHφ1: free_evars φ1^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: ∀ (ψ : Pattern) (x : evar),
  free_evars φ2^[[evar:x↦ψ]]
  ⊆ free_evars φ2 ∪ free_evars ψ
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ specialize (IHφ2 ψ x').
Σ: Signature
φ1, φ2, ψ: Pattern
x': evar
IHφ1: free_evars φ1^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars ψ
IHφ2: free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ2 ∪ free_evars ψ
free_evars φ1^[[evar:x'↦ψ]]
∪ free_evars φ2^[[evar:x'↦ψ]]
⊆ free_evars φ1 ∪ free_evars φ2 ∪ free_evars ψ
  set_solver.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (x : evar),
free_evars φ^[[evar:x↦ψ]] ⊆ free_evars φ ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ^[[evar:x'↦ψ]]
⊆ free_evars φ ∪ free_evars ψ apply IHφ.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (x : evar),
free_evars φ^[[evar:x↦ψ]] ⊆ free_evars φ ∪ free_evars ψ
ψ: Pattern
x': evar
free_evars φ^[[evar:x'↦ψ]]
⊆ free_evars φ ∪ free_evars ψ apply IHφ.
Qed.


Lemma bound_to_free_variable_subst :
forall φ x m n ψ,
  m > n ->
  well_formed_closed_ex_aux ψ 0 ->
  well_formed_closed_ex_aux φ m -> x ∉ free_evars φ ->
  bevar_subst φ ψ n = free_evar_subst (evar_open n x φ) ψ x.
Σ: Signature
∀ (φ : Pattern) (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ m
      → x ∉ free_evars φ
        → φ^[evar:n↦ψ] =
          (evar_open n x φ)^[[evar:x↦ψ]]
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ m
      → x ∉ free_evars φ
        → φ^[evar:n↦ψ] =
          (evar_open n x φ)^[[evar:x↦ψ]]
induction φ; intros x' m n' ψ H WFψ H0 H1; cbn; auto.
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ free_evars (patt_free_evar x)
patt_free_evar x =
(if decide (x' = x) then ψ else patt_free_evar x)
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_bound_evar n) m
H1: x' ∉ free_evars (patt_bound_evar n)
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x'
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[[evar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) m
H1: x' ∉ free_evars (patt_app φ1 φ2)
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_evars (patt_imp φ1 φ2)
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ m
      → x ∉ free_evars φ
        → φ^[evar:n↦ψ] =
          (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_exists φ) m
H1: x' ∉ free_evars (patt_exists φ)
patt_exists φ^[evar:S n'↦ψ] =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ m
      → x ∉ free_evars φ
        → φ^[evar:n↦ψ] =
          (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_mu φ) m
H1: x' ∉ free_evars (patt_mu φ)
patt_mu φ^[evar:n'↦ψ] =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
-
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ free_evars (patt_free_evar x)
patt_free_evar x =
(if decide (x' = x) then ψ else patt_free_evar x) destruct (decide (x' = x)); simpl.
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ free_evars (patt_free_evar x)
e: x' = x
patt_free_evar x = ψ
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ free_evars (patt_free_evar x)
n: x' ≠ x
patt_free_evar x = patt_free_evar x
  +
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ free_evars (patt_free_evar x)
e: x' = x
patt_free_evar x = ψ simpl in H1.
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ {[x]}
e: x' = x
patt_free_evar x = ψ apply not_elem_of_singleton_1 in H1.
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ≠ x
e: x' = x
patt_free_evar x = ψ congruence.
  +
Σ: Signature
x, x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_free_evar x) m
H1: x' ∉ free_evars (patt_free_evar x)
n: x' ≠ x
patt_free_evar x = patt_free_evar x reflexivity.
-
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_bound_evar n) m
H1: x' ∉ free_evars (patt_bound_evar n)
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end =
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x'
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[[evar:x'↦ψ]] case_match; auto.
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_bound_evar n) m
H1: x' ∉ free_evars (patt_bound_evar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
ψ = (patt_free_evar x')^[[evar:x'↦ψ]] simpl.
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_bound_evar n) m
H1: x' ∉ free_evars (patt_bound_evar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
ψ =
(if decide (x' = x') then ψ else patt_free_evar x') case_match; auto; simpl in H0; case_match; auto.
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
l: n < m
H4: decide (n < m) = left l
H0: true
H1: x' ∉ free_evars (patt_bound_evar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
n0: x' ≠ x'
H3: decide (x' = x') = right n0
ψ = patt_free_evar x'
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
n1: ¬ n < m
H4: decide (n < m) = right n1
H0: false
H1: x' ∉ free_evars (patt_bound_evar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
n0: x' ≠ x'
H3: decide (x' = x') = right n0
ψ = patt_free_evar x'
  contradiction.
Σ: Signature
n: db_index
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
n1: ¬ n < m
H4: decide (n < m) = right n1
H0: false
H1: x' ∉ free_evars (patt_bound_evar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
n0: x' ≠ x'
H3: decide (x' = x') = right n0
ψ = patt_free_evar x' lia.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) m
H1: x' ∉ free_evars (patt_app φ1 φ2)
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] simpl in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) m
H1: x' ∉ free_evars φ1 ∪ free_evars φ2
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] apply not_elem_of_union in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) m
H1: (x' ∉ free_evars φ1) ∧ x' ∉ free_evars φ2
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] destruct H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_app φ1 φ2) m
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m &&
well_formed_closed_ex_aux φ2 m
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  apply andb_true_iff in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
∧ well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_app φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  erewrite -> IHφ1, -> IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_app (evar_open n' ?x φ1)^[[evar:?x↦ψ]]
  (evar_open n' ?x0 φ2)^[[evar:?x0↦ψ]] =
patt_app φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?x0 ∉ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?x ∉ free_evars φ1 reflexivity.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
x' ∉ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
x' ∉ free_evars φ1 all: eassumption.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_evars (patt_imp φ1 φ2)
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] simpl in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_evars φ1 ∪ free_evars φ2
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] apply not_elem_of_union in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
H1: (x' ∉ free_evars φ1) ∧ x' ∉ free_evars φ2
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] destruct H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m &&
well_formed_closed_ex_aux φ2 m
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  apply andb_true_iff in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
∧ well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_imp φ1^[evar:n'↦ψ] φ2^[evar:n'↦ψ] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  erewrite -> IHφ1, -> IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
patt_imp (evar_open n' ?x φ1)^[[evar:?x↦ψ]]
  (evar_open n' ?x0 φ2)^[[evar:?x0↦ψ]] =
patt_imp φ1^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
  φ2^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?x0 ∉ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?x ∉ free_evars φ1 reflexivity.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
x' ∉ free_evars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
well_formed_closed_ex_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ1 m
      → x ∉ free_evars φ1
        → φ1^[evar:n↦ψ] =
          (evar_open n x φ1)^[[evar:x↦ψ]]
IHφ2: ∀ (x : evar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_ex_aux ψ 0
    → well_formed_closed_ex_aux φ2 m
      → x ∉ free_evars φ2
        → φ2^[evar:n↦ψ] =
          (evar_open n x φ2)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ1 m = true
H3: well_formed_closed_ex_aux φ2 m = true
H1: x' ∉ free_evars φ1
H2: x' ∉ free_evars φ2
x' ∉ free_evars φ1 all: eassumption.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_exists φ) m
H1: x' ∉ free_evars (patt_exists φ)
patt_exists φ^[evar:S n'↦ψ] =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:x'↦ψ]] simpl in H0, H1.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
patt_exists φ^[evar:S n'↦ψ] =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:x'↦ψ]] erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
patt_exists (evar_open (S n') ?x φ)^[[evar:?x↦ψ]] =
patt_exists
  φ^[evar:S n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
?m > S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
?x ∉ free_evars φ reflexivity.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
?m > S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
x' ∉ free_evars φ instantiate (1 := S m).
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
S m > S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux φ (S m)
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
x' ∉ free_evars φ 
  all: try eassumption.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ (S m)
H1: x' ∉ free_evars φ
S m > S n' lia.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux (patt_mu φ) m
H1: x' ∉ free_evars (patt_mu φ)
patt_mu φ^[evar:n'↦ψ] =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] simpl in H0, H1.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
patt_mu φ^[evar:n'↦ψ] =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]] erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
patt_mu (evar_open n' ?x φ)^[[evar:?x↦ψ]] =
patt_mu φ^[evar:n'↦patt_free_evar x']^[[evar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
?m > n'
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
?x ∉ free_evars φ reflexivity.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
?m > n'
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
well_formed_closed_ex_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_ex_aux ψ 0
→ well_formed_closed_ex_aux φ m
→ x ∉ free_evars φ → φ^[evar:n↦ψ] = (evar_open n x φ)^[[evar:x↦ψ]]
x': evar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_ex_aux ψ 0
H0: well_formed_closed_ex_aux φ m
H1: x' ∉ free_evars φ
x' ∉ free_evars φ all: eassumption.
Qed.

Lemma bound_to_free_set_variable_subst :
forall φ X m n ψ,
  m > n ->
  well_formed_closed_mu_aux ψ 0 ->
  well_formed_closed_mu_aux φ m -> X ∉ free_svars φ ->
  bsvar_subst φ ψ n = free_svar_subst (svar_open n X φ) ψ X.
Σ: Signature
∀ (φ : Pattern) (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ m
      → X ∉ free_svars φ
        → φ^[svar:n↦ψ] =
          (svar_open n X φ)^[[svar:X↦ψ]]
Proof.
Σ: Signature
∀ (φ : Pattern) (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ m
      → X ∉ free_svars φ
        → φ^[svar:n↦ψ] =
          (svar_open n X φ)^[[svar:X↦ψ]]
induction φ; intros x' m n' ψ H WFψ H0 H1; cbn; auto.
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ free_svars (patt_free_svar x)
patt_free_svar x =
(if decide (x' = x) then ψ else patt_free_svar x)
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_bound_svar n) m
H1: x' ∉ free_svars (patt_bound_svar n)
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end =
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar x'
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[[svar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) m
H1: x' ∉ free_svars (patt_app φ1 φ2)
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_svars (patt_imp φ1 φ2)
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ m
      → X ∉ free_svars φ
        → φ^[svar:n↦ψ] =
          (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_exists φ) m
H1: x' ∉ free_svars (patt_exists φ)
patt_exists φ^[svar:n'↦ψ] =
patt_exists
  φ^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ m
      → X ∉ free_svars φ
        → φ^[svar:n↦ψ] =
          (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_mu φ) m
H1: x' ∉ free_svars (patt_mu φ)
patt_mu φ^[svar:S n'↦ψ] =
patt_mu φ^[svar:S n'↦patt_free_svar x']^[[svar:x'↦ψ]]
-
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ free_svars (patt_free_svar x)
patt_free_svar x =
(if decide (x' = x) then ψ else patt_free_svar x) destruct (decide (x' = x)); simpl.
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ free_svars (patt_free_svar x)
e: x' = x
patt_free_svar x = ψ
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ free_svars (patt_free_svar x)
n: x' ≠ x
patt_free_svar x = patt_free_svar x
  +
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ free_svars (patt_free_svar x)
e: x' = x
patt_free_svar x = ψ simpl in H1.
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ {[x]}
e: x' = x
patt_free_svar x = ψ apply not_elem_of_singleton_1 in H1.
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ≠ x
e: x' = x
patt_free_svar x = ψ congruence.
  +
Σ: Signature
x, x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_free_svar x) m
H1: x' ∉ free_svars (patt_free_svar x)
n: x' ≠ x
patt_free_svar x = patt_free_svar x reflexivity.
-
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_bound_svar n) m
H1: x' ∉ free_svars (patt_bound_svar n)
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => ψ
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end =
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar x'
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end^[[svar:x'↦ψ]] case_match; auto.
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_bound_svar n) m
H1: x' ∉ free_svars (patt_bound_svar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
ψ = (patt_free_svar x')^[[svar:x'↦ψ]] simpl.
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_bound_svar n) m
H1: x' ∉ free_svars (patt_bound_svar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
ψ =
(if decide (x' = x') then ψ else patt_free_svar x') case_match; auto; simpl in H0; case_match; auto.
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
l: n < m
H4: decide (n < m) = left l
H0: true
H1: x' ∉ free_svars (patt_bound_svar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
n0: x' ≠ x'
H3: decide (x' = x') = right n0
ψ = patt_free_svar x'
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
n1: ¬ n < m
H4: decide (n < m) = right n1
H0: false
H1: x' ∉ free_svars (patt_bound_svar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
n0: x' ≠ x'
H3: decide (x' = x') = right n0
ψ = patt_free_svar x'
  contradiction.
Σ: Signature
n: db_index
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
n1: ¬ n < m
H4: decide (n < m) = right n1
H0: false
H1: x' ∉ free_svars (patt_bound_svar n)
e: n = n'
H2: compare_nat n n' = Nat_equal n n' e
n0: x' ≠ x'
H3: decide (x' = x') = right n0
ψ = patt_free_svar x' lia.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) m
H1: x' ∉ free_svars (patt_app φ1 φ2)
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] simpl in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) m
H1: x' ∉ free_svars φ1 ∪ free_svars φ2
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] apply not_elem_of_union in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) m
H1: (x' ∉ free_svars φ1) ∧ x' ∉ free_svars φ2
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] destruct H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_app φ1 φ2) m
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  apply andb_true_iff in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
∧ well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_app φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  erewrite -> IHφ1, -> IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_app (svar_open n' ?X φ1)^[[svar:?X↦ψ]]
  (svar_open n' ?X0 φ2)^[[svar:?X0↦ψ]] =
patt_app φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?X0 ∉ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?X ∉ free_svars φ1 reflexivity.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
x' ∉ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
x' ∉ free_svars φ1 all: eassumption.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_svars (patt_imp φ1 φ2)
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] simpl in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_svars φ1 ∪ free_svars φ2
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] apply not_elem_of_union in H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
H1: (x' ∉ free_svars φ1) ∧ x' ∉ free_svars φ2
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] destruct H1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] simpl in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  apply andb_true_iff in H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
∧ well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] destruct H0.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_imp φ1^[svar:n'↦ψ] φ2^[svar:n'↦ψ] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  erewrite -> IHφ1, -> IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
patt_imp (svar_open n' ?X φ1)^[[svar:?X↦ψ]]
  (svar_open n' ?X0 φ2)^[[svar:?X0↦ψ]] =
patt_imp φ1^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
  φ2^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?X0 ∉ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?X ∉ free_svars φ1 reflexivity.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m0 > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ2 ?m0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
x' ∉ free_svars φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
?m > n'
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
well_formed_closed_mu_aux φ1 ?m
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ1 m
      → X ∉ free_svars φ1
        → φ1^[svar:n↦ψ] =
          (svar_open n X φ1)^[[svar:X↦ψ]]
IHφ2: ∀ (X : svar) (m n : nat) (ψ : Pattern),
  m > n
  → well_formed_closed_mu_aux ψ 0
    → well_formed_closed_mu_aux φ2 m
      → X ∉ free_svars φ2
        → φ2^[svar:n↦ψ] =
          (svar_open n X φ2)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ1 m = true
H3: well_formed_closed_mu_aux φ2 m = true
H1: x' ∉ free_svars φ1
H2: x' ∉ free_svars φ2
x' ∉ free_svars φ1 all: eassumption.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_exists φ) m
H1: x' ∉ free_svars (patt_exists φ)
patt_exists φ^[svar:n'↦ψ] =
patt_exists
  φ^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] simpl in H0, H1.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
patt_exists φ^[svar:n'↦ψ] =
patt_exists
  φ^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]] erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
patt_exists (svar_open n' ?X φ)^[[svar:?X↦ψ]] =
patt_exists
  φ^[svar:n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
?m > n'
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
?X ∉ free_svars φ reflexivity.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
?m > n'
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ m
H1: x' ∉ free_svars φ
x' ∉ free_svars φ all: eassumption.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux (patt_mu φ) m
H1: x' ∉ free_svars (patt_mu φ)
patt_mu φ^[svar:S n'↦ψ] =
patt_mu φ^[svar:S n'↦patt_free_svar x']^[[svar:x'↦ψ]] simpl in H0, H1.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
patt_mu φ^[svar:S n'↦ψ] =
patt_mu φ^[svar:S n'↦patt_free_svar x']^[[svar:x'↦ψ]] erewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
patt_mu (svar_open (S n') ?X φ)^[[svar:?X↦ψ]] =
patt_mu φ^[svar:S n'↦patt_free_svar x']^[[svar:x'↦ψ]]
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
?m > S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
?X ∉ free_svars φ reflexivity.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
?m > S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux φ ?m
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
x' ∉ free_svars φ instantiate (1 := S m).
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
S m > S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux ψ 0
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
well_formed_closed_mu_aux φ (S m)
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
x' ∉ free_svars φ 
  all: try eassumption.
Σ: Signature
φ: Pattern
IHφ: ∀ (X : svar) (m n : nat) (ψ : Pattern),
m > n
→ well_formed_closed_mu_aux ψ 0
→ well_formed_closed_mu_aux φ m
→ X ∉ free_svars φ → φ^[svar:n↦ψ] = (svar_open n X φ)^[[svar:X↦ψ]]
x': svar
m, n': nat
ψ: Pattern
H: m > n'
WFψ: well_formed_closed_mu_aux ψ 0
H0: well_formed_closed_mu_aux φ (S m)
H1: x' ∉ free_svars φ
S m > S n' lia.
Qed.

Lemma evar_open_no_negative_occurrence :
forall φ db1 db2 x,
  (no_negative_occurrence_db_b db1 (evar_open db2 x φ) ->
  no_negative_occurrence_db_b db1 φ) /\
  (no_positive_occurrence_db_b db1 (evar_open db2 x φ) ->
  no_positive_occurrence_db_b db1 φ).
Σ: Signature
∀ (φ : Pattern) (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 (evar_open db2 x φ)
   → no_negative_occurrence_db_b db1 φ)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ)
     → no_positive_occurrence_db_b db1 φ)
Proof.
Σ: Signature
∀ (φ : Pattern) (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 (evar_open db2 x φ)
   → no_negative_occurrence_db_b db1 φ)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ)
     → no_positive_occurrence_db_b db1 φ)
induction φ; intros db1 db2 x'; simpl; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (evar_open db2 x' (patt_app φ1 φ2))
 → no_negative_occurrence_db_b db1 (patt_app φ1 φ2))
∧ (no_positive_occurrence_db_b db1
     (evar_open db2 x' (patt_app φ1 φ2))
   → no_positive_occurrence_db_b db1 (patt_app φ1 φ2))
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (evar_open db2 x' (patt_imp φ1 φ2))
 → no_negative_occurrence_db_b db1 (patt_imp φ1 φ2))
∧ (no_positive_occurrence_db_b db1
     (evar_open db2 x' (patt_imp φ1 φ2))
   → no_positive_occurrence_db_b db1 (patt_imp φ1 φ2))
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ)
   → no_negative_occurrence_db_b db1 φ)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ)
     → no_positive_occurrence_db_b db1 φ)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (evar_open db2 x' (patt_exists φ))
 → no_negative_occurrence_db_b db1 (patt_exists φ))
∧ (no_positive_occurrence_db_b db1
     (evar_open db2 x' (patt_exists φ))
   → no_positive_occurrence_db_b db1 (patt_exists φ))
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ)
   → no_negative_occurrence_db_b db1 φ)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ)
     → no_positive_occurrence_db_b db1 φ)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (evar_open db2 x' (patt_mu φ))
 → no_negative_occurrence_db_b db1 (patt_mu φ))
∧ (no_positive_occurrence_db_b db1
     (evar_open db2 x' (patt_mu φ))
   → no_positive_occurrence_db_b db1 (patt_mu φ))
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (evar_open db2 x' (patt_app φ1 φ2))
 → no_negative_occurrence_db_b db1 (patt_app φ1 φ2))
∧ (no_positive_occurrence_db_b db1
     (evar_open db2 x' (patt_app φ1 φ2))
   → no_positive_occurrence_db_b db1 (patt_app φ1 φ2)) split; intros.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1
  (evar_open db2 x' (patt_app φ1 φ2))
no_negative_occurrence_db_b db1 (patt_app φ1 φ2)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1
  (evar_open db2 x' (patt_app φ1 φ2))
no_positive_occurrence_db_b db1 (patt_app φ1 φ2)
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1
  (evar_open db2 x' (patt_app φ1 φ2))
no_negative_occurrence_db_b db1 (patt_app φ1 φ2) apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  ((fix bevar_subst
      (phi psi : Pattern) 
      (x : db_index) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst phi' psi (S x))
      | patt_mu phi' =>
          patt_mu (bevar_subst phi' psi x)
      end) φ1 (patt_free_evar x') db2) = true
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  ((fix bevar_subst
      (phi psi : Pattern) 
      (x : db_index) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst phi' psi (S x))
      | patt_mu phi' =>
          patt_mu (bevar_subst phi' psi x)
      end) φ2 (patt_free_evar x') db2) = true
no_negative_occurrence_db_b db1 (patt_app φ1 φ2)
    apply IHφ1 in E1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_negative_occurrence_db_b db1 φ1
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  ((fix bevar_subst
      (phi psi : Pattern) 
      (x : db_index) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst phi' psi (S x))
      | patt_mu phi' =>
          patt_mu (bevar_subst phi' psi x)
      end) φ2 (patt_free_evar x') db2) = true
no_negative_occurrence_db_b db1 (patt_app φ1 φ2) apply IHφ2 in E2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_negative_occurrence_db_b db1 φ1
E2: no_negative_occurrence_db_b db1 φ2
no_negative_occurrence_db_b db1 (patt_app φ1 φ2)
    cbn.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_negative_occurrence_db_b db1 φ1
E2: no_negative_occurrence_db_b db1 φ2
no_negative_occurrence_db_b db1 φ1 &&
no_negative_occurrence_db_b db1 φ2
    now rewrite -> E1, -> E2.
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1
  (evar_open db2 x' (patt_app φ1 φ2))
no_positive_occurrence_db_b db1 (patt_app φ1 φ2) apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  ((fix bevar_subst
      (phi psi : Pattern) 
      (x : db_index) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst phi' psi (S x))
      | patt_mu phi' =>
          patt_mu (bevar_subst phi' psi x)
      end) φ1 (patt_free_evar x') db2) = true
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  ((fix bevar_subst
      (phi psi : Pattern) 
      (x : db_index) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst phi' psi (S x))
      | patt_mu phi' =>
          patt_mu (bevar_subst phi' psi x)
      end) φ2 (patt_free_evar x') db2) = true
no_positive_occurrence_db_b db1 (patt_app φ1 φ2)
    apply IHφ1 in E1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_positive_occurrence_db_b db1 φ1
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  ((fix bevar_subst
      (phi psi : Pattern) 
      (x : db_index) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst phi' psi (S x))
      | patt_mu phi' =>
          patt_mu (bevar_subst phi' psi x)
      end) φ2 (patt_free_evar x') db2) = true
no_positive_occurrence_db_b db1 (patt_app φ1 φ2) apply IHφ2 in E2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_positive_occurrence_db_b db1 φ1
E2: no_positive_occurrence_db_b db1 φ2
no_positive_occurrence_db_b db1 (patt_app φ1 φ2)
    cbn.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_positive_occurrence_db_b db1 φ1
E2: no_positive_occurrence_db_b db1 φ2
no_positive_occurrence_db_b db1 φ1 &&
no_positive_occurrence_db_b db1 φ2
    now rewrite -> E1, -> E2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (evar_open db2 x' (patt_imp φ1 φ2))
 → no_negative_occurrence_db_b db1 (patt_imp φ1 φ2))
∧ (no_positive_occurrence_db_b db1
     (evar_open db2 x' (patt_imp φ1 φ2))
   → no_positive_occurrence_db_b db1 (patt_imp φ1 φ2)) split; intros.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1
  (evar_open db2 x' (patt_imp φ1 φ2))
no_negative_occurrence_db_b db1 (patt_imp φ1 φ2)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1
  (evar_open db2 x' (patt_imp φ1 φ2))
no_positive_occurrence_db_b db1 (patt_imp φ1 φ2)
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_negative_occurrence_db_b db1
  (evar_open db2 x' (patt_imp φ1 φ2))
no_negative_occurrence_db_b db1 (patt_imp φ1 φ2) apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  ((fix bevar_subst
      (phi psi : Pattern) 
      (x : db_index) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst phi' psi (S x))
      | patt_mu phi' =>
          patt_mu (bevar_subst phi' psi x)
      end) φ1 (patt_free_evar x') db2) = true
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  ((fix bevar_subst
      (phi psi : Pattern) 
      (x : db_index) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst phi' psi (S x))
      | patt_mu phi' =>
          patt_mu (bevar_subst phi' psi x)
      end) φ2 (patt_free_evar x') db2) = true
no_negative_occurrence_db_b db1 (patt_imp φ1 φ2)
    apply IHφ1 in E1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_positive_occurrence_db_b db1 φ1
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  ((fix bevar_subst
      (phi psi : Pattern) 
      (x : db_index) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst phi' psi (S x))
      | patt_mu phi' =>
          patt_mu (bevar_subst phi' psi x)
      end) φ2 (patt_free_evar x') db2) = true
no_negative_occurrence_db_b db1 (patt_imp φ1 φ2) apply IHφ2 in E2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_positive_occurrence_db_b db1 φ1
E2: no_negative_occurrence_db_b db1 φ2
no_negative_occurrence_db_b db1 (patt_imp φ1 φ2)
    cbn.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_positive_occurrence_db_b db1 φ1
E2: no_negative_occurrence_db_b db1 φ2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1 φ1 &&
no_negative_occurrence_db_b db1 φ2
    now rewrite -> E1, -> E2.
  -
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
H: no_positive_occurrence_db_b db1
  (evar_open db2 x' (patt_imp φ1 φ2))
no_positive_occurrence_db_b db1 (patt_imp φ1 φ2) apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  ((fix bevar_subst
      (phi psi : Pattern) 
      (x : db_index) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst phi' psi (S x))
      | patt_mu phi' =>
          patt_mu (bevar_subst phi' psi x)
      end) φ1 (patt_free_evar x') db2) = true
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  ((fix bevar_subst
      (phi psi : Pattern) 
      (x : db_index) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst phi' psi (S x))
      | patt_mu phi' =>
          patt_mu (bevar_subst phi' psi x)
      end) φ2 (patt_free_evar x') db2) = true
no_positive_occurrence_db_b db1 (patt_imp φ1 φ2)
    apply IHφ1 in E1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_negative_occurrence_db_b db1 φ1
E2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  ((fix bevar_subst
      (phi psi : Pattern) 
      (x : db_index) {struct phi} : Pattern :=
      match phi with
      | patt_free_evar x' => patt_free_evar x'
      | patt_free_svar x' => patt_free_svar x'
      | patt_bound_evar n =>
          match compare_nat n x with
          | Nat_less _ _ _ => patt_bound_evar n
          | Nat_equal _ _ _ => psi
          | Nat_greater _ _ _ =>
              patt_bound_evar (Nat.pred n)
          end
      | patt_bound_svar n => patt_bound_svar n
      | patt_sym sigma => patt_sym sigma
      | patt_app phi1 phi2 =>
          patt_app (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_bott => patt_bott
      | patt_imp phi1 phi2 =>
          patt_imp (bevar_subst phi1 psi x)
            (bevar_subst phi2 psi x)
      | patt_exists phi' =>
          patt_exists
            (bevar_subst phi' psi (S x))
      | patt_mu phi' =>
          patt_mu (bevar_subst phi' psi x)
      end) φ2 (patt_free_evar x') db2) = true
no_positive_occurrence_db_b db1 (patt_imp φ1 φ2) apply IHφ2 in E2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_negative_occurrence_db_b db1 φ1
E2: no_positive_occurrence_db_b db1 φ2
no_positive_occurrence_db_b db1 (patt_imp φ1 φ2)
    cbn.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ1)
   → no_negative_occurrence_db_b db1 φ1)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ1)
     → no_positive_occurrence_db_b db1 φ1)
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1
     (evar_open db2 x φ2)
   → no_negative_occurrence_db_b db1 φ2)
  ∧ (no_positive_occurrence_db_b db1
       (evar_open db2 x φ2)
     → no_positive_occurrence_db_b db1 φ2)
db1, db2: db_index
x': evar
E1: no_negative_occurrence_db_b db1 φ1
E2: no_positive_occurrence_db_b db1 φ2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1 φ1 &&
no_positive_occurrence_db_b db1 φ2 now rewrite -> E1, -> E2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 (evar_open db2 x φ)
→ no_negative_occurrence_db_b db1 φ)
∧ (no_positive_occurrence_db_b db1 (evar_open db2 x φ)
→ no_positive_occurrence_db_b db1 φ)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (evar_open db2 x' (patt_exists φ))
 → no_negative_occurrence_db_b db1 (patt_exists φ))
∧ (no_positive_occurrence_db_b db1
     (evar_open db2 x' (patt_exists φ))
   → no_positive_occurrence_db_b db1 (patt_exists φ)) cbn.
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 (evar_open db2 x φ)
→ no_negative_occurrence_db_b db1 φ)
∧ (no_positive_occurrence_db_b db1 (evar_open db2 x φ)
→ no_positive_occurrence_db_b db1 φ)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   φ^[evar:S db2↦patt_free_evar x']
 → no_negative_occurrence_db_b db1 φ)
∧ (no_positive_occurrence_db_b db1
     φ^[evar:S db2↦patt_free_evar x']
   → no_positive_occurrence_db_b db1 φ) split; intros; eapply IHφ; eassumption.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 (evar_open db2 x φ)
→ no_negative_occurrence_db_b db1 φ)
∧ (no_positive_occurrence_db_b db1 (evar_open db2 x φ)
→ no_positive_occurrence_db_b db1 φ)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b db1
   (evar_open db2 x' (patt_mu φ))
 → no_negative_occurrence_db_b db1 (patt_mu φ))
∧ (no_positive_occurrence_db_b db1
     (evar_open db2 x' (patt_mu φ))
   → no_positive_occurrence_db_b db1 (patt_mu φ)) cbn.
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 (evar_open db2 x φ)
→ no_negative_occurrence_db_b db1 φ)
∧ (no_positive_occurrence_db_b db1 (evar_open db2 x φ)
→ no_positive_occurrence_db_b db1 φ)
db1, db2: db_index
x': evar
(no_negative_occurrence_db_b (S db1)
   φ^[evar:db2↦patt_free_evar x']
 → no_negative_occurrence_db_b (S db1) φ)
∧ (no_positive_occurrence_db_b (S db1)
     φ^[evar:db2↦patt_free_evar x']
   → no_positive_occurrence_db_b (S db1) φ) split; intros; eapply IHφ; eassumption.
Qed.

Lemma evar_open_positive : forall φ n x,
well_formed_positive (evar_open n x φ) = true ->
well_formed_positive φ = true.
Σ: Signature
∀ (φ : Pattern) (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ) = true
  → well_formed_positive φ = true
Proof.
Σ: Signature
∀ (φ : Pattern) (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ) = true
  → well_formed_positive φ = true
induction φ; intros n' x' H; cbn; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ1) =
  true → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ2) =
  true → well_formed_positive φ2 = true
n': db_index
x': evar
H: well_formed_positive
  (evar_open n' x' (patt_app φ1 φ2)) = true
well_formed_positive φ1 && well_formed_positive φ2 =
true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ1) =
  true → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ2) =
  true → well_formed_positive φ2 = true
n': db_index
x': evar
H: well_formed_positive
  (evar_open n' x' (patt_imp φ1 φ2)) = true
well_formed_positive φ1 && well_formed_positive φ2 =
true
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ) = true
  → well_formed_positive φ = true
n': db_index
x': evar
H: well_formed_positive
  (evar_open n' x' (patt_exists φ)) = true
well_formed_positive φ = true
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ) = true
  → well_formed_positive φ = true
n': db_index
x': evar
H: well_formed_positive (evar_open n' x' (patt_mu φ)) =
true
no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ = true
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ1) =
  true → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ2) =
  true → well_formed_positive φ2 = true
n': db_index
x': evar
H: well_formed_positive
  (evar_open n' x' (patt_app φ1 φ2)) = true
well_formed_positive φ1 && well_formed_positive φ2 =
true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ1) =
  true → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ2) =
  true → well_formed_positive φ2 = true
n': db_index
x': evar
H: well_formed_positive
  φ1^[evar:n'↦patt_free_evar x'] &&
well_formed_positive
  φ2^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ1 && well_formed_positive φ2 =
true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ1) =
  true → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ2) =
  true → well_formed_positive φ2 = true
n': db_index
x': evar
E1: well_formed_positive
  φ1^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ2^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ1 && well_formed_positive φ2 =
true
  erewrite -> IHφ1, -> IHφ2; eauto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ1) =
  true → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ2) =
  true → well_formed_positive φ2 = true
n': db_index
x': evar
H: well_formed_positive
  (evar_open n' x' (patt_imp φ1 φ2)) = true
well_formed_positive φ1 && well_formed_positive φ2 =
true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ1) =
  true → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ2) =
  true → well_formed_positive φ2 = true
n': db_index
x': evar
H: well_formed_positive
  φ1^[evar:n'↦patt_free_evar x'] &&
well_formed_positive
  φ2^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ1 && well_formed_positive φ2 =
true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ1) =
  true → well_formed_positive φ1 = true
IHφ2: ∀ (n : db_index) (x : evar),
  well_formed_positive (evar_open n x φ2) =
  true → well_formed_positive φ2 = true
n': db_index
x': evar
E1: well_formed_positive
  φ1^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ2^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ1 && well_formed_positive φ2 =
true
  erewrite -> IHφ1, -> IHφ2; eauto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive (evar_open n x φ) = true
→ well_formed_positive φ = true
n': db_index
x': evar
H: well_formed_positive
  (evar_open n' x' (patt_exists φ)) = true
well_formed_positive φ = true simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive (evar_open n x φ) = true
→ well_formed_positive φ = true
n': db_index
x': evar
H: well_formed_positive
  φ^[evar:S n'↦patt_free_evar x'] = true
well_formed_positive φ = true eapply IHφ; eauto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive (evar_open n x φ) = true
→ well_formed_positive φ = true
n': db_index
x': evar
H: well_formed_positive (evar_open n' x' (patt_mu φ)) =
true
no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ = true simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive (evar_open n x φ) = true
→ well_formed_positive φ = true
n': db_index
x': evar
H: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] &&
well_formed_positive φ^[evar:n'↦patt_free_evar x'] =
true
no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive (evar_open n x φ) = true
→ well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ = true
  apply andb_true_iff.
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive (evar_open n x φ) = true
→ well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
no_negative_occurrence_db_b 0 φ = true
∧ well_formed_positive φ = true split.
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive (evar_open n x φ) = true
→ well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
no_negative_occurrence_db_b 0 φ = true
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive (evar_open n x φ) = true
→ well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ = true
  eapply evar_open_no_negative_occurrence.
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive (evar_open n x φ) = true
→ well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
no_negative_occurrence_db_b 0 (evar_open ?db2 ?x φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive (evar_open n x φ) = true
→ well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ = true eassumption.
Σ: Signature
φ: Pattern
IHφ: ∀ (n : db_index) (x : evar),
well_formed_positive (evar_open n x φ) = true
→ well_formed_positive φ = true
n': db_index
x': evar
E1: no_negative_occurrence_db_b 0
  φ^[evar:n'↦patt_free_evar x'] = true
E2: well_formed_positive
  φ^[evar:n'↦patt_free_evar x'] = true
well_formed_positive φ = true
  eapply IHφ; eauto.
Qed.

Lemma bevar_subst_closed_mu :
forall φ ψ n m,
well_formed_closed_mu_aux φ m = true ->
well_formed_closed_mu_aux ψ m = true
->
well_formed_closed_mu_aux (bevar_subst φ ψ n) m = true.
Σ: Signature
∀ (φ ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
induction φ; intros ψ n' m H H0; cbn; auto.
Σ: Signature
n: db_index
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_bound_evar n) m =
true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end m = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_app φ1 φ2) m =
true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_imp φ1 φ2) m =
true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_mu φ) m = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ^[evar:n'↦ψ] (S m) = true
*
Σ: Signature
n: db_index
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_bound_evar n) m =
true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end m = true break_match_goal; simpl in H0, H; simpl; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_app φ1 φ2) m =
true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
E1: well_formed_closed_mu_aux φ1 m = true
E2: well_formed_closed_mu_aux φ2 m = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true erewrite IHφ1, IHφ2; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_imp φ1 φ2) m =
true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ1 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ1^[evar:n↦ψ] m =
      true
IHφ2: ∀ (ψ : Pattern) (n m : db_index),
  well_formed_closed_mu_aux φ2 m = true
  → well_formed_closed_mu_aux ψ m = true
    → well_formed_closed_mu_aux φ2^[evar:n↦ψ] m =
      true
ψ: Pattern
n', m: db_index
E1: well_formed_closed_mu_aux φ1 m = true
E2: well_formed_closed_mu_aux φ2 m = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ1^[evar:n'↦ψ] m &&
well_formed_closed_mu_aux φ2^[evar:n'↦ψ] m = true erewrite IHφ1, IHφ2; auto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
well_formed_closed_mu_aux φ m = true
→ well_formed_closed_mu_aux ψ m = true
→ well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux (patt_mu φ) m = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ^[evar:n'↦ψ] (S m) = true simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
well_formed_closed_mu_aux φ m = true
→ well_formed_closed_mu_aux ψ m = true
→ well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ (S m) = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux φ^[evar:n'↦ψ] (S m) = true rewrite -> IHφ; auto.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
well_formed_closed_mu_aux φ m = true
→ well_formed_closed_mu_aux ψ m = true
→ well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ (S m) = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux ψ (S m) = true eapply well_formed_closed_mu_aux_ind.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
well_formed_closed_mu_aux φ m = true
→ well_formed_closed_mu_aux ψ m = true
→ well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ (S m) = true
H0: well_formed_closed_mu_aux ψ m = true
?ind_svar1 ≤ S m
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
well_formed_closed_mu_aux φ m = true
→ well_formed_closed_mu_aux ψ m = true
→ well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ (S m) = true
H0: well_formed_closed_mu_aux ψ m = true
well_formed_closed_mu_aux ψ ?ind_svar1 = true
  2: eassumption.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n m : db_index),
well_formed_closed_mu_aux φ m = true
→ well_formed_closed_mu_aux ψ m = true
→ well_formed_closed_mu_aux φ^[evar:n↦ψ] m = true
ψ: Pattern
n', m: db_index
H: well_formed_closed_mu_aux φ (S m) = true
H0: well_formed_closed_mu_aux ψ m = true
m ≤ S m lia.
Qed.

Lemma bevar_subst_closed_ex :
forall φ ψ n,
well_formed_closed_ex_aux φ (S n) = true ->
well_formed_closed_ex_aux ψ n = true
->
well_formed_closed_ex_aux (bevar_subst φ ψ n) n = true.
Σ: Signature
∀ (φ ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
induction φ; intros ψ n' H H0; cbn; auto.
Σ: Signature
n: db_index
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_bound_evar n)
  (S n') = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n' = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_app φ1 φ2) (S n') =
true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_imp φ1 φ2) (S n') =
true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_exists φ) (S n') =
true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ^[evar:S n'↦ψ] (S n') =
true
*
Σ: Signature
n: db_index
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_bound_evar n)
  (S n') = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end n' = true break_match_goal; simpl in H0, H; simpl; auto.
Σ: Signature
n: db_index
ψ: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_ex_aux ψ n' = true
l: n < n'
Heqc: compare_nat n n' = Nat_less n n' l
(if decide (n < n') then true else false) = true
Σ: Signature
n: db_index
ψ: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_ex_aux ψ n' = true
g: n > n'
Heqc: compare_nat n n' = Nat_greater n n' g
(if decide (Nat.pred n < n') then true else false) =
true
  repeat case_match; auto.
Σ: Signature
n: db_index
ψ: Pattern
n': nat
H: (if decide (n < S n') then true else false) = true
H0: well_formed_closed_ex_aux ψ n' = true
g: n > n'
Heqc: compare_nat n n' = Nat_greater n n' g
(if decide (Nat.pred n < n') then true else false) =
true do 2 case_match; auto; lia.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_app φ1 φ2) (S n') =
true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ1 (S n') &&
well_formed_closed_ex_aux φ2 (S n') = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
E1: well_formed_closed_ex_aux φ1 (S n') = true
E2: well_formed_closed_ex_aux φ2 (S n') = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true erewrite IHφ1, IHφ2; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_imp φ1 φ2) (S n') =
true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ1 (S n') &&
well_formed_closed_ex_aux φ2 (S n') = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ1 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ1^[evar:n↦ψ] n =
      true
IHφ2: ∀ (ψ : Pattern) (n : nat),
  well_formed_closed_ex_aux φ2 (S n) = true
  → well_formed_closed_ex_aux ψ n = true
    → well_formed_closed_ex_aux φ2^[evar:n↦ψ] n =
      true
ψ: Pattern
n': nat
E1: well_formed_closed_ex_aux φ1 (S n') = true
E2: well_formed_closed_ex_aux φ2 (S n') = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ1^[evar:n'↦ψ] n' &&
well_formed_closed_ex_aux φ2^[evar:n'↦ψ] n' = true erewrite IHφ1, IHφ2; auto.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
well_formed_closed_ex_aux φ (S n) = true
→ well_formed_closed_ex_aux ψ n = true
→ well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux (patt_exists φ) (S n') =
true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ^[evar:S n'↦ψ] (S n') =
true simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
well_formed_closed_ex_aux φ (S n) = true
→ well_formed_closed_ex_aux ψ n = true
→ well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ (S (S n')) = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux φ^[evar:S n'↦ψ] (S n') =
true rewrite -> IHφ; auto.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
well_formed_closed_ex_aux φ (S n) = true
→ well_formed_closed_ex_aux ψ n = true
→ well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ (S (S n')) = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux ψ (S n') = true eapply well_formed_closed_ex_aux_ind.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
well_formed_closed_ex_aux φ (S n) = true
→ well_formed_closed_ex_aux ψ n = true
→ well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ (S (S n')) = true
H0: well_formed_closed_ex_aux ψ n' = true
?ind_evar1 ≤ S n'
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
well_formed_closed_ex_aux φ (S n) = true
→ well_formed_closed_ex_aux ψ n = true
→ well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ (S (S n')) = true
H0: well_formed_closed_ex_aux ψ n' = true
well_formed_closed_ex_aux ψ ?ind_evar1 = true
  2: eassumption.
Σ: Signature
φ: Pattern
IHφ: ∀ (ψ : Pattern) (n : nat),
well_formed_closed_ex_aux φ (S n) = true
→ well_formed_closed_ex_aux ψ n = true
→ well_formed_closed_ex_aux φ^[evar:n↦ψ] n = true
ψ: Pattern
n': nat
H: well_formed_closed_ex_aux φ (S (S n')) = true
H0: well_formed_closed_ex_aux ψ n' = true
n' ≤ S n' lia.
Qed.

Lemma bevar_subst_positive :
forall φ ψ n, mu_free φ ->
well_formed_positive φ = true -> well_formed_positive ψ = true
->
well_formed_positive (bevar_subst φ ψ n) = true.
Σ: Signature
∀ (φ ψ : Pattern) (n : db_index),
  mu_free φ
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
Proof.
Σ: Signature
∀ (φ ψ : Pattern) (n : db_index),
  mu_free φ
  → well_formed_positive φ = true
    → well_formed_positive ψ = true
      → well_formed_positive φ^[evar:n↦ψ] = true
induction φ; intros ψ n' H H0 H1; cbn; auto.
Σ: Signature
n: db_index
ψ: Pattern
n': db_index
H: mu_free (patt_bound_evar n)
H0: well_formed_positive (patt_bound_evar n) = true
H1: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H: mu_free (patt_app φ1 φ2)
H0: well_formed_positive (patt_app φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H: mu_free (patt_imp φ1 φ2)
H0: well_formed_positive (patt_imp φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
*
Σ: Signature
n: db_index
ψ: Pattern
n': db_index
H: mu_free (patt_bound_evar n)
H0: well_formed_positive (patt_bound_evar n) = true
H1: well_formed_positive ψ = true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => ψ
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
  end = true break_match_goal; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H: mu_free (patt_app φ1 φ2)
H0: well_formed_positive (patt_app φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H: mu_free φ1 && mu_free φ2
H0: well_formed_positive (patt_app φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
E1: mu_free φ1 = true
E2: mu_free φ2 = true
H0: well_formed_positive (patt_app φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
  apply andb_true_iff in H0 as [E1' E2'].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
E1: mu_free φ1 = true
E2: mu_free φ2 = true
E1': (fix well_formed_positive 
 (phi : Pattern) : bool :=
   match phi with
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_positive psi1 &&
       well_formed_positive psi2
   | patt_exists psi => well_formed_positive psi
   | patt_mu psi =>
       no_negative_occurrence_db_b 0 psi &&
       well_formed_positive psi
   | _ => true
   end) φ1 = true
E2': (fix well_formed_positive 
 (phi : Pattern) : bool :=
   match phi with
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_positive psi1 &&
       well_formed_positive psi2
   | patt_exists psi => well_formed_positive psi
   | patt_mu psi =>
       no_negative_occurrence_db_b 0 psi &&
       well_formed_positive psi
   | _ => true
   end) φ2 = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
  rewrite -> IHφ1, -> IHφ2; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H: mu_free (patt_imp φ1 φ2)
H0: well_formed_positive (patt_imp φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
H: mu_free φ1 && mu_free φ2
H0: well_formed_positive (patt_imp φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
E1: mu_free φ1 = true
E2: mu_free φ2 = true
H0: well_formed_positive (patt_imp φ1 φ2) = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
  apply andb_true_iff in H0 as [E1' E2'].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ1
  → well_formed_positive φ1 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ1^[evar:n↦ψ] =
        true
IHφ2: ∀ (ψ : Pattern) (n : db_index),
  mu_free φ2
  → well_formed_positive φ2 = true
    → well_formed_positive ψ = true
      → well_formed_positive φ2^[evar:n↦ψ] =
        true
ψ: Pattern
n': db_index
E1: mu_free φ1 = true
E2: mu_free φ2 = true
E1': (fix well_formed_positive 
 (phi : Pattern) : bool :=
   match phi with
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_positive psi1 &&
       well_formed_positive psi2
   | patt_exists psi => well_formed_positive psi
   | patt_mu psi =>
       no_negative_occurrence_db_b 0 psi &&
       well_formed_positive psi
   | _ => true
   end) φ1 = true
E2': (fix well_formed_positive 
 (phi : Pattern) : bool :=
   match phi with
   | patt_app psi1 psi2 | patt_imp psi1 psi2 =>
       well_formed_positive psi1 &&
       well_formed_positive psi2
   | patt_exists psi => well_formed_positive psi
   | patt_mu psi =>
       no_negative_occurrence_db_b 0 psi &&
       well_formed_positive psi
   | _ => true
   end) φ2 = true
H1: well_formed_positive ψ = true
well_formed_positive φ1^[evar:n'↦ψ] &&
well_formed_positive φ2^[evar:n'↦ψ] = true
  now rewrite -> IHφ1, -> IHφ2.
Qed.

Theorem evar_quantify_closed_ex :
forall φ x n, well_formed_closed_ex_aux φ n ->
well_formed_closed_ex_aux (evar_quantify x n φ) (S n) = true.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ n
  → well_formed_closed_ex_aux (evar_quantify x n φ)
      (S n) = true
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ n
  → well_formed_closed_ex_aux (evar_quantify x n φ)
      (S n) = true
induction φ; intros x' n' H; cbn; auto.
Σ: Signature
x, x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x) n'
well_formed_closed_ex_aux
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) (S n') = true
Σ: Signature
n: db_index
x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_bound_evar n) n'
(if decide (n < S n') then true else false) = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ1) 
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ2) 
      (S n) = true
x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_app φ1 φ2) n'
well_formed_closed_ex_aux (evar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_ex_aux (evar_quantify x' n' φ2)
  (S n') = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ1) 
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ2) 
      (S n) = true
x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_imp φ1 φ2) n'
well_formed_closed_ex_aux 
  (evar_quantify x' n' φ1) 
  (S n') &&
well_formed_closed_ex_aux 
  (evar_quantify x' n' φ2) 
  (S n') = true
*
Σ: Signature
x, x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x) n'
well_formed_closed_ex_aux
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) (S n') = true destruct (decide (x' = x)); simpl; auto.
Σ: Signature
x, x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_free_evar x) n'
e: x' = x
(if decide (n' < S n') then true else false) = true
  case_match; try lia.
*
Σ: Signature
n: db_index
x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_bound_evar n) n'
(if decide (n < S n') then true else false) = true simpl in H.
Σ: Signature
n: db_index
x': evar
n': db_index
H: if decide (n < n') then true else false
(if decide (n < S n') then true else false) = true repeat case_match; auto; lia.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ1) 
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ2) 
      (S n) = true
x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_app φ1 φ2) n'
well_formed_closed_ex_aux (evar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_ex_aux (evar_quantify x' n' φ2)
  (S n') = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ1) 
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ2) 
      (S n) = true
x': evar
n': db_index
H: well_formed_closed_ex_aux φ1 n' &&
well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux (evar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_ex_aux (evar_quantify x' n' φ2)
  (S n') = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ1) 
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ2) 
      (S n) = true
x': evar
n': db_index
E1: well_formed_closed_ex_aux φ1 n' = true
E2: well_formed_closed_ex_aux φ2 n' = true
well_formed_closed_ex_aux (evar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_ex_aux (evar_quantify x' n' φ2)
  (S n') = true now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ1) 
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ2) 
      (S n) = true
x': evar
n': db_index
H: well_formed_closed_ex_aux (patt_imp φ1 φ2) n'
well_formed_closed_ex_aux (evar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_ex_aux (evar_quantify x' n' φ2)
  (S n') = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ1) 
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ2) 
      (S n) = true
x': evar
n': db_index
H: well_formed_closed_ex_aux φ1 n' &&
well_formed_closed_ex_aux φ2 n'
well_formed_closed_ex_aux (evar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_ex_aux (evar_quantify x' n' φ2)
  (S n') = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ1 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ1) 
      (S n) = true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_closed_ex_aux φ2 n
  → well_formed_closed_ex_aux
      (evar_quantify x n φ2) 
      (S n) = true
x': evar
n': db_index
E1: well_formed_closed_ex_aux φ1 n' = true
E2: well_formed_closed_ex_aux φ2 n' = true
well_formed_closed_ex_aux (evar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_ex_aux (evar_quantify x' n' φ2)
  (S n') = true now rewrite -> IHφ1, -> IHφ2. 
Qed.

Theorem svar_quantify_closed_mu :
forall φ X n, well_formed_closed_mu_aux φ n ->
well_formed_closed_mu_aux (svar_quantify X n φ) (S n) = true.
Σ: Signature
∀ (φ : Pattern) (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ n
  → well_formed_closed_mu_aux (svar_quantify X n φ)
      (S n) = true
Proof.
Σ: Signature
∀ (φ : Pattern) (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ n
  → well_formed_closed_mu_aux (svar_quantify X n φ)
      (S n) = true
induction φ; intros x' n' H; cbn; auto.
Σ: Signature
x, x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
well_formed_closed_mu_aux
  (if decide (x' = x)
   then patt_bound_svar n'
   else patt_free_svar x) (S n') = true
Σ: Signature
n: db_index
x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_bound_svar n) n'
(if decide (n < S n') then true else false) = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ1) 
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ2) 
      (S n) = true
x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_app φ1 φ2) n'
well_formed_closed_mu_aux (svar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_mu_aux (svar_quantify x' n' φ2)
  (S n') = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ1) 
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ2) 
      (S n) = true
x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_imp φ1 φ2) n'
well_formed_closed_mu_aux 
  (svar_quantify x' n' φ1) 
  (S n') &&
well_formed_closed_mu_aux 
  (svar_quantify x' n' φ2) 
  (S n') = true
*
Σ: Signature
x, x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
well_formed_closed_mu_aux
  (if decide (x' = x)
   then patt_bound_svar n'
   else patt_free_svar x) (S n') = true destruct (decide (x' = x)); simpl; auto.
Σ: Signature
x, x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_free_svar x) n'
e: x' = x
(if decide (n' < S n') then true else false) = true
  case_match; try lia.
*
Σ: Signature
n: db_index
x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_bound_svar n) n'
(if decide (n < S n') then true else false) = true simpl in H.
Σ: Signature
n: db_index
x': svar
n': db_index
H: if decide (n < n') then true else false
(if decide (n < S n') then true else false) = true repeat case_match; auto; lia.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ1) 
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ2) 
      (S n) = true
x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_app φ1 φ2) n'
well_formed_closed_mu_aux (svar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_mu_aux (svar_quantify x' n' φ2)
  (S n') = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ1) 
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ2) 
      (S n) = true
x': svar
n': db_index
H: well_formed_closed_mu_aux φ1 n' &&
well_formed_closed_mu_aux φ2 n'
well_formed_closed_mu_aux (svar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_mu_aux (svar_quantify x' n' φ2)
  (S n') = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ1) 
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ2) 
      (S n) = true
x': svar
n': db_index
E1: well_formed_closed_mu_aux φ1 n' = true
E2: well_formed_closed_mu_aux φ2 n' = true
well_formed_closed_mu_aux (svar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_mu_aux (svar_quantify x' n' φ2)
  (S n') = true now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ1) 
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ2) 
      (S n) = true
x': svar
n': db_index
H: well_formed_closed_mu_aux (patt_imp φ1 φ2) n'
well_formed_closed_mu_aux (svar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_mu_aux (svar_quantify x' n' φ2)
  (S n') = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ1) 
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ2) 
      (S n) = true
x': svar
n': db_index
H: well_formed_closed_mu_aux φ1 n' &&
well_formed_closed_mu_aux φ2 n'
well_formed_closed_mu_aux (svar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_mu_aux (svar_quantify x' n' φ2)
  (S n') = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ1 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ1) 
      (S n) = true
IHφ2: ∀ (X : svar) (n : db_index),
  well_formed_closed_mu_aux φ2 n
  → well_formed_closed_mu_aux
      (svar_quantify X n φ2) 
      (S n) = true
x': svar
n': db_index
E1: well_formed_closed_mu_aux φ1 n' = true
E2: well_formed_closed_mu_aux φ2 n' = true
well_formed_closed_mu_aux (svar_quantify x' n' φ1)
  (S n') &&
well_formed_closed_mu_aux (svar_quantify x' n' φ2)
  (S n') = true now rewrite -> IHφ1, -> IHφ2. 
Qed.

Theorem evar_quantify_closed_mu :
forall φ x n m, well_formed_closed_mu_aux φ m ->
well_formed_closed_mu_aux (evar_quantify x n φ) m = true.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ m
  → well_formed_closed_mu_aux (evar_quantify x n φ) m =
    true
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ m
  → well_formed_closed_mu_aux (evar_quantify x n φ) m =
    true
induction φ; intros x' n' m H; cbn; auto.
Σ: Signature
x, x': evar
n', m: db_index
H: well_formed_closed_mu_aux (patt_free_evar x) m
well_formed_closed_mu_aux
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) m = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux (patt_app φ1 φ2) m
well_formed_closed_mu_aux (evar_quantify x' n' φ1) m &&
well_formed_closed_mu_aux (evar_quantify x' n' φ2) m =
true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
well_formed_closed_mu_aux (evar_quantify x' n' φ1) m &&
well_formed_closed_mu_aux (evar_quantify x' n' φ2) m =
true
-
Σ: Signature
x, x': evar
n', m: db_index
H: well_formed_closed_mu_aux (patt_free_evar x) m
well_formed_closed_mu_aux
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) m = true destruct (decide (x' = x)); simpl; auto.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux (patt_app φ1 φ2) m
well_formed_closed_mu_aux (evar_quantify x' n' φ1) m &&
well_formed_closed_mu_aux (evar_quantify x' n' φ2) m =
true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m
well_formed_closed_mu_aux (evar_quantify x' n' φ1) m &&
well_formed_closed_mu_aux (evar_quantify x' n' φ2) m =
true repeat case_match; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m
well_formed_closed_mu_aux (evar_quantify x' n' φ1) m &&
well_formed_closed_mu_aux (evar_quantify x' n' φ2) m =
true
  destruct_and!.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux (evar_quantify x' n' φ1) m &&
well_formed_closed_mu_aux (evar_quantify x' n' φ2) m =
true split_and!.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux (evar_quantify x' n' φ1) m =
true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux (evar_quantify x' n' φ2) m =
true
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux (evar_quantify x' n' φ1) m =
true apply IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux φ1 m assumption.
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux (evar_quantify x' n' φ2) m =
true apply IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H0: well_formed_closed_mu_aux φ1 m = true
H1: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux φ2 m assumption.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux (patt_imp φ1 φ2) m
well_formed_closed_mu_aux (evar_quantify x' n' φ1) m &&
well_formed_closed_mu_aux (evar_quantify x' n' φ2) m =
true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
H: well_formed_closed_mu_aux φ1 m &&
well_formed_closed_mu_aux φ2 m
well_formed_closed_mu_aux (evar_quantify x' n' φ1) m &&
well_formed_closed_mu_aux (evar_quantify x' n' φ2) m =
true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ1 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ1) m = true
IHφ2: ∀ (x : evar) (n m : db_index),
  well_formed_closed_mu_aux φ2 m
  → well_formed_closed_mu_aux
      (evar_quantify x n φ2) m = true
x': evar
n', m: db_index
E1: well_formed_closed_mu_aux φ1 m = true
E2: well_formed_closed_mu_aux φ2 m = true
well_formed_closed_mu_aux (evar_quantify x' n' φ1) m &&
well_formed_closed_mu_aux (evar_quantify x' n' φ2) m =
true now rewrite -> IHφ1, -> IHφ2.
Qed.

Theorem svar_quantify_closed_ex :
forall φ X n m, well_formed_closed_ex_aux φ m ->
well_formed_closed_ex_aux (svar_quantify X n φ) m = true.
Σ: Signature
∀ (φ : Pattern) (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ m
  → well_formed_closed_ex_aux (svar_quantify X n φ) m =
    true
Proof.
Σ: Signature
∀ (φ : Pattern) (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ m
  → well_formed_closed_ex_aux (svar_quantify X n φ) m =
    true
induction φ; intros x' n' m H; cbn; auto.
Σ: Signature
x, x': svar
n', m: db_index
H: well_formed_closed_ex_aux (patt_free_svar x) m
well_formed_closed_ex_aux
  (if decide (x' = x)
   then patt_bound_svar n'
   else patt_free_svar x) m = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux (patt_app φ1 φ2) m
well_formed_closed_ex_aux (svar_quantify x' n' φ1) m &&
well_formed_closed_ex_aux (svar_quantify x' n' φ2) m =
true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
well_formed_closed_ex_aux (svar_quantify x' n' φ1) m &&
well_formed_closed_ex_aux (svar_quantify x' n' φ2) m =
true
-
Σ: Signature
x, x': svar
n', m: db_index
H: well_formed_closed_ex_aux (patt_free_svar x) m
well_formed_closed_ex_aux
  (if decide (x' = x)
   then patt_bound_svar n'
   else patt_free_svar x) m = true destruct (decide (x' = x)); simpl; auto.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux (patt_app φ1 φ2) m
well_formed_closed_ex_aux (svar_quantify x' n' φ1) m &&
well_formed_closed_ex_aux (svar_quantify x' n' φ2) m =
true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux φ1 m &&
well_formed_closed_ex_aux φ2 m
well_formed_closed_ex_aux (svar_quantify x' n' φ1) m &&
well_formed_closed_ex_aux (svar_quantify x' n' φ2) m =
true repeat case_match; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux φ1 m &&
well_formed_closed_ex_aux φ2 m
well_formed_closed_ex_aux (svar_quantify x' n' φ1) m &&
well_formed_closed_ex_aux (svar_quantify x' n' φ2) m =
true
  destruct_and!.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux (svar_quantify x' n' φ1) m &&
well_formed_closed_ex_aux (svar_quantify x' n' φ2) m =
true split_and!.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux (svar_quantify x' n' φ1) m =
true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux (svar_quantify x' n' φ2) m =
true
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux (svar_quantify x' n' φ1) m =
true apply IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux φ1 m assumption.
  +
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux (svar_quantify x' n' φ2) m =
true apply IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H0: well_formed_closed_ex_aux φ1 m = true
H1: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux φ2 m assumption.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux (patt_imp φ1 φ2) m
well_formed_closed_ex_aux (svar_quantify x' n' φ1) m &&
well_formed_closed_ex_aux (svar_quantify x' n' φ2) m =
true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
H: well_formed_closed_ex_aux φ1 m &&
well_formed_closed_ex_aux φ2 m
well_formed_closed_ex_aux (svar_quantify x' n' φ1) m &&
well_formed_closed_ex_aux (svar_quantify x' n' φ2) m =
true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ1 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ1) m = true
IHφ2: ∀ (X : svar) (n m : db_index),
  well_formed_closed_ex_aux φ2 m
  → well_formed_closed_ex_aux
      (svar_quantify X n φ2) m = true
x': svar
n', m: db_index
E1: well_formed_closed_ex_aux φ1 m = true
E2: well_formed_closed_ex_aux φ2 m = true
well_formed_closed_ex_aux (svar_quantify x' n' φ1) m &&
well_formed_closed_ex_aux (svar_quantify x' n' φ2) m =
true now rewrite -> IHφ1, -> IHφ2.
Qed.

Theorem no_occ_quantify : 
∀ (φ : Pattern) (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ
 → no_negative_occurrence_db_b db1 (evar_quantify x db2 φ))
∧ (no_positive_occurrence_db_b db1 φ
   → no_positive_occurrence_db_b db1 (evar_quantify x db2 φ)).
Σ: Signature
∀ (φ : Pattern) (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ))
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ))
Proof.
Σ: Signature
∀ (φ : Pattern) (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ))
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ))
induction φ; split; intros H; cbn; auto.
Σ: Signature
x: evar
db1, db2: db_index
x0: evar
H: no_negative_occurrence_db_b db1 (patt_free_evar x)
no_negative_occurrence_db_b db1
  (if decide (x0 = x)
   then patt_bound_evar db2
   else patt_free_evar x)
Σ: Signature
x: evar
db1, db2: db_index
x0: evar
H: no_positive_occurrence_db_b db1 (patt_free_evar x)
no_positive_occurrence_db_b db1
  (if decide (x0 = x)
   then patt_bound_evar db2
   else patt_free_evar x)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ1))
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ1))
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ2))
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ2))
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_app φ1 φ2)
no_negative_occurrence_db_b db1
  (evar_quantify x db2 φ1) &&
no_negative_occurrence_db_b db1
  (evar_quantify x db2 φ2)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ1))
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ1))
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ2))
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ2))
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_app φ1 φ2)
no_positive_occurrence_db_b db1
  (evar_quantify x db2 φ1) &&
no_positive_occurrence_db_b db1
  (evar_quantify x db2 φ2)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ1))
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ1))
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ2))
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ2))
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_imp φ1 φ2)
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  (evar_quantify x db2 φ1) &&
no_negative_occurrence_db_b db1
  (evar_quantify x db2 φ2)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ1))
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ1))
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ2))
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ2))
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_imp φ1 φ2)
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  (evar_quantify x db2 φ1) &&
no_positive_occurrence_db_b db1
  (evar_quantify x db2 φ2)
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ))
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ))
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_exists φ)
no_negative_occurrence_db_b db1
  (evar_quantify x (S db2) φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ))
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ))
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_exists φ)
no_positive_occurrence_db_b db1
  (evar_quantify x (S db2) φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ))
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ))
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_mu φ)
no_negative_occurrence_db_b 
  (S db1) (evar_quantify x db2 φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ))
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ))
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_mu φ)
no_positive_occurrence_db_b 
  (S db1) (evar_quantify x db2 φ)
1-2: destruct (decide (x0 = x)); simpl; auto.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ1))
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ1))
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ2))
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ2))
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_app φ1 φ2)
no_negative_occurrence_db_b db1
  (evar_quantify x db2 φ1) &&
no_negative_occurrence_db_b db1
  (evar_quantify x db2 φ2)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ1))
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ1))
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ2))
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ2))
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_app φ1 φ2)
no_positive_occurrence_db_b db1
  (evar_quantify x db2 φ1) &&
no_positive_occurrence_db_b db1
  (evar_quantify x db2 φ2)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ1))
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ1))
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ2))
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ2))
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_imp φ1 φ2)
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) db1
  (evar_quantify x db2 φ1) &&
no_negative_occurrence_db_b db1
  (evar_quantify x db2 φ2)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ1
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ1))
  ∧ (no_positive_occurrence_db_b db1 φ1
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ1))
IHφ2: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ2
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ2))
  ∧ (no_positive_occurrence_db_b db1 φ2
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ2))
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_imp φ1 φ2)
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) db1
  (evar_quantify x db2 φ1) &&
no_positive_occurrence_db_b db1
  (evar_quantify x db2 φ2)
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ))
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ))
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_exists φ)
no_negative_occurrence_db_b db1
  (evar_quantify x (S db2) φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ))
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ))
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_exists φ)
no_positive_occurrence_db_b db1
  (evar_quantify x (S db2) φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ))
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ))
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_mu φ)
no_negative_occurrence_db_b 
  (S db1) (evar_quantify x db2 φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
  (no_negative_occurrence_db_b db1 φ
   → no_negative_occurrence_db_b db1
       (evar_quantify x db2 φ))
  ∧ (no_positive_occurrence_db_b db1 φ
     → no_positive_occurrence_db_b db1
         (evar_quantify x db2 φ))
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_mu φ)
no_positive_occurrence_db_b 
  (S db1) (evar_quantify x db2 φ)
1-4: simpl in H; apply andb_true_iff in H as [E1 E2];
     specialize (IHφ1 db1 db2 x) as [IH1 IH2];
     specialize (IHφ2 db1 db2 x) as [IH1' IH2'];
     try rewrite -> IH1; try rewrite -> IH1'; 
     try rewrite -> IH2; try rewrite -> IH2'; auto.
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ
→ no_negative_occurrence_db_b db1 (evar_quantify x db2 φ))
∧ (no_positive_occurrence_db_b db1 φ
→ no_positive_occurrence_db_b db1 (evar_quantify x db2 φ))
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_exists φ)
no_negative_occurrence_db_b db1
  (evar_quantify x (S db2) φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ
→ no_negative_occurrence_db_b db1 (evar_quantify x db2 φ))
∧ (no_positive_occurrence_db_b db1 φ
→ no_positive_occurrence_db_b db1 (evar_quantify x db2 φ))
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_exists φ)
no_positive_occurrence_db_b db1
  (evar_quantify x (S db2) φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ
→ no_negative_occurrence_db_b db1 (evar_quantify x db2 φ))
∧ (no_positive_occurrence_db_b db1 φ
→ no_positive_occurrence_db_b db1 (evar_quantify x db2 φ))
db1, db2: db_index
x: evar
H: no_negative_occurrence_db_b db1 (patt_mu φ)
no_negative_occurrence_db_b (S db1)
  (evar_quantify x db2 φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (db1 db2 : db_index) (x : evar),
(no_negative_occurrence_db_b db1 φ
→ no_negative_occurrence_db_b db1 (evar_quantify x db2 φ))
∧ (no_positive_occurrence_db_b db1 φ
→ no_positive_occurrence_db_b db1 (evar_quantify x db2 φ))
db1, db2: db_index
x: evar
H: no_positive_occurrence_db_b db1 (patt_mu φ)
no_positive_occurrence_db_b (S db1)
  (evar_quantify x db2 φ)
1-4: simpl in H; now apply IHφ.
Qed.


Theorem evar_quantify_positive :
forall φ x n, well_formed_positive φ ->
well_formed_positive (evar_quantify x n φ) = true.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  well_formed_positive φ
  → well_formed_positive (evar_quantify x n φ) = true
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  well_formed_positive φ
  → well_formed_positive (evar_quantify x n φ) = true
induction φ; intros x' n' H; cbn; auto.
Σ: Signature
x, x': evar
n': db_index
H: well_formed_positive (patt_free_evar x)
well_formed_positive
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive (evar_quantify x n φ1) =
    true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive (evar_quantify x n φ2) =
    true
x': evar
n': db_index
H: well_formed_positive (patt_app φ1 φ2)
well_formed_positive (evar_quantify x' n' φ1) &&
well_formed_positive (evar_quantify x' n' φ2) = true
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive (evar_quantify x n φ1) =
    true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive (evar_quantify x n φ2) =
    true
x': evar
n': db_index
H: well_formed_positive (patt_imp φ1 φ2)
well_formed_positive (evar_quantify x' n' φ1) &&
well_formed_positive (evar_quantify x' n' φ2) = true
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  well_formed_positive φ
  → well_formed_positive (evar_quantify x n φ) =
    true
x': evar
n': db_index
H: well_formed_positive (patt_mu φ)
no_negative_occurrence_db_b 0 (evar_quantify x' n' φ) &&
well_formed_positive (evar_quantify x' n' φ) = true
*
Σ: Signature
x, x': evar
n': db_index
H: well_formed_positive (patt_free_evar x)
well_formed_positive
  (if decide (x' = x)
   then patt_bound_evar n'
   else patt_free_evar x) = true destruct (decide (x' = x)); simpl; auto.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive (evar_quantify x n φ1) =
    true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive (evar_quantify x n φ2) =
    true
x': evar
n': db_index
H: well_formed_positive (patt_app φ1 φ2)
well_formed_positive (evar_quantify x' n' φ1) &&
well_formed_positive (evar_quantify x' n' φ2) = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive (evar_quantify x n φ1) =
    true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive (evar_quantify x n φ2) =
    true
x': evar
n': db_index
H: well_formed_positive φ1 && well_formed_positive φ2
well_formed_positive (evar_quantify x' n' φ1) &&
well_formed_positive (evar_quantify x' n' φ2) = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive (evar_quantify x n φ1) =
    true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive (evar_quantify x n φ2) =
    true
x': evar
n': db_index
E1: well_formed_positive φ1 = true
E2: well_formed_positive φ2 = true
well_formed_positive (evar_quantify x' n' φ1) &&
well_formed_positive (evar_quantify x' n' φ2) = true now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive (evar_quantify x n φ1) =
    true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive (evar_quantify x n φ2) =
    true
x': evar
n': db_index
H: well_formed_positive (patt_imp φ1 φ2)
well_formed_positive (evar_quantify x' n' φ1) &&
well_formed_positive (evar_quantify x' n' φ2) = true simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive (evar_quantify x n φ1) =
    true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive (evar_quantify x n φ2) =
    true
x': evar
n': db_index
H: well_formed_positive φ1 && well_formed_positive φ2
well_formed_positive (evar_quantify x' n' φ1) &&
well_formed_positive (evar_quantify x' n' φ2) = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  well_formed_positive φ1
  → well_formed_positive (evar_quantify x n φ1) =
    true
IHφ2: ∀ (x : evar) (n : db_index),
  well_formed_positive φ2
  → well_formed_positive (evar_quantify x n φ2) =
    true
x': evar
n': db_index
E1: well_formed_positive φ1 = true
E2: well_formed_positive φ2 = true
well_formed_positive (evar_quantify x' n' φ1) &&
well_formed_positive (evar_quantify x' n' φ2) = true now rewrite -> IHφ1, -> IHφ2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive (evar_quantify x n φ) = true
x': evar
n': db_index
H: well_formed_positive (patt_mu φ)
no_negative_occurrence_db_b 0 (evar_quantify x' n' φ) &&
well_formed_positive (evar_quantify x' n' φ) = true simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive (evar_quantify x n φ) = true
x': evar
n': db_index
H: no_negative_occurrence_db_b 0 φ &&
well_formed_positive φ
no_negative_occurrence_db_b 0 (evar_quantify x' n' φ) &&
well_formed_positive (evar_quantify x' n' φ) = true apply andb_true_iff in H as [E1 E2].
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive (evar_quantify x n φ) = true
x': evar
n': db_index
E1: no_negative_occurrence_db_b 0 φ = true
E2: well_formed_positive φ = true
no_negative_occurrence_db_b 0 (evar_quantify x' n' φ) &&
well_formed_positive (evar_quantify x' n' φ) = true apply andb_true_iff.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive (evar_quantify x n φ) = true
x': evar
n': db_index
E1: no_negative_occurrence_db_b 0 φ = true
E2: well_formed_positive φ = true
no_negative_occurrence_db_b 0 (evar_quantify x' n' φ) =
true
∧ well_formed_positive (evar_quantify x' n' φ) = true split.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive (evar_quantify x n φ) = true
x': evar
n': db_index
E1: no_negative_occurrence_db_b 0 φ = true
E2: well_formed_positive φ = true
no_negative_occurrence_db_b 0 (evar_quantify x' n' φ) =
true
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive (evar_quantify x n φ) = true
x': evar
n': db_index
E1: no_negative_occurrence_db_b 0 φ = true
E2: well_formed_positive φ = true
well_formed_positive (evar_quantify x' n' φ) = true
  -
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive (evar_quantify x n φ) = true
x': evar
n': db_index
E1: no_negative_occurrence_db_b 0 φ = true
E2: well_formed_positive φ = true
no_negative_occurrence_db_b 0 (evar_quantify x' n' φ) =
true now apply no_occ_quantify.
  -
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
well_formed_positive φ → well_formed_positive (evar_quantify x n φ) = true
x': evar
n': db_index
E1: no_negative_occurrence_db_b 0 φ = true
E2: well_formed_positive φ = true
well_formed_positive (evar_quantify x' n' φ) = true now apply IHφ.
Qed.

Corollary evar_quantify_well_formed :
forall φ x, well_formed φ ->
  well_formed (patt_exists (evar_quantify x 0 φ)) = true.
Σ: Signature
∀ (φ : Pattern) (x : evar),
  well_formed φ
  → well_formed (patt_exists (evar_quantify x 0 φ)) =
    true
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar),
  well_formed φ
  → well_formed (patt_exists (evar_quantify x 0 φ)) =
    true
intros φ x H.
Σ: Signature
φ: Pattern
x: evar
H: well_formed φ
well_formed (patt_exists (evar_quantify x 0 φ)) = true
unfold well_formed, well_formed_closed in *.
Σ: Signature
φ: Pattern
x: evar
H: [&& well_formed_positive φ,
    well_formed_closed_mu_aux φ 0
  & well_formed_closed_ex_aux φ 0]
[&& well_formed_positive
      (patt_exists (evar_quantify x 0 φ)),
    well_formed_closed_mu_aux
      (patt_exists (evar_quantify x 0 φ)) 0
  & well_formed_closed_ex_aux
      (patt_exists (evar_quantify x 0 φ)) 0] = true
destruct_and!.
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
[&& well_formed_positive
      (patt_exists (evar_quantify x 0 φ)),
    well_formed_closed_mu_aux
      (patt_exists (evar_quantify x 0 φ)) 0
  & well_formed_closed_ex_aux
      (patt_exists (evar_quantify x 0 φ)) 0] = true
split_and!; simpl.
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_positive (evar_quantify x 0 φ) = true
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_closed_mu_aux (evar_quantify x 0 φ) 0 =
true
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_closed_ex_aux (evar_quantify x 0 φ) 1 =
true
-
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_positive (evar_quantify x 0 φ) = true apply evar_quantify_positive.
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_positive φ assumption.
-
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_closed_mu_aux (evar_quantify x 0 φ) 0 =
true apply evar_quantify_closed_mu.
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_closed_mu_aux φ 0 assumption.
-
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_closed_ex_aux (evar_quantify x 0 φ) 1 =
true apply evar_quantify_closed_ex.
Σ: Signature
φ: Pattern
x: evar
H0: well_formed_positive φ = true
H: well_formed_closed_mu_aux φ 0 = true
H2: well_formed_closed_ex_aux φ 0 = true
well_formed_closed_ex_aux φ 0 assumption.
Qed.

Theorem evar_quantify_not_free :
forall φ x n, x ∉ (free_evars (evar_quantify x n φ)).
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ)
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ)
induction φ; intros x' n'; simpl.
Σ: Signature
x, x': evar
n': db_index
x'
∉ free_evars
    (if decide (x' = x)
     then patt_bound_evar n'
     else patt_free_evar x)
Σ: Signature
x: svar
x': evar
n': db_index
x' ∉ ∅
Σ: Signature
n: db_index
x': evar
n': db_index
x' ∉ ∅
Σ: Signature
n: db_index
x': evar
n': db_index
x' ∉ ∅
Σ: Signature
sigma: symbols
x': evar
n': db_index
x' ∉ ∅
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
x'
∉ free_evars (evar_quantify x' n' φ1)
  ∪ free_evars (evar_quantify x' n' φ2)
Σ: Signature
x': evar
n': db_index
x' ∉ ∅
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
x'
∉ free_evars (evar_quantify x' n' φ1)
  ∪ free_evars (evar_quantify x' n' φ2)
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ)
x': evar
n': db_index
x' ∉ free_evars (evar_quantify x' (S n') φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ)
x': evar
n': db_index
x' ∉ free_evars (evar_quantify x' n' φ)
2-5, 7: apply not_elem_of_empty.
Σ: Signature
x, x': evar
n': db_index
x'
∉ free_evars
    (if decide (x' = x)
     then patt_bound_evar n'
     else patt_free_evar x)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
x'
∉ free_evars (evar_quantify x' n' φ1)
  ∪ free_evars (evar_quantify x' n' φ2)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
x'
∉ free_evars (evar_quantify x' n' φ1)
  ∪ free_evars (evar_quantify x' n' φ2)
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ)
x': evar
n': db_index
x' ∉ free_evars (evar_quantify x' (S n') φ)
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ)
x': evar
n': db_index
x' ∉ free_evars (evar_quantify x' n' φ)
*
Σ: Signature
x, x': evar
n': db_index
x'
∉ free_evars
    (if decide (x' = x)
     then patt_bound_evar n'
     else patt_free_evar x) destruct (decide (x' = x)); simpl.
Σ: Signature
x, x': evar
n': db_index
e: x' = x
x' ∉ ∅
Σ: Signature
x, x': evar
n': db_index
n: x' ≠ x
x' ∉ {[x]}
  -
Σ: Signature
x, x': evar
n': db_index
e: x' = x
x' ∉ ∅ apply not_elem_of_empty.
  -
Σ: Signature
x, x': evar
n': db_index
n: x' ≠ x
x' ∉ {[x]} subst.
Σ: Signature
x, x': evar
n': db_index
n: x' ≠ x
x' ∉ {[x]} now apply not_elem_of_singleton_2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
x'
∉ free_evars (evar_quantify x' n' φ1)
  ∪ free_evars (evar_quantify x' n' φ2) apply not_elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
(x' ∉ free_evars (evar_quantify x' n' φ1))
∧ x' ∉ free_evars (evar_quantify x' n' φ2) split.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
x' ∉ free_evars (evar_quantify x' n' φ1)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
x' ∉ free_evars (evar_quantify x' n' φ2) apply IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
x' ∉ free_evars (evar_quantify x' n' φ2) apply IHφ2.
*
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
x'
∉ free_evars (evar_quantify x' n' φ1)
  ∪ free_evars (evar_quantify x' n' φ2) apply not_elem_of_union.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
(x' ∉ free_evars (evar_quantify x' n' φ1))
∧ x' ∉ free_evars (evar_quantify x' n' φ2) split.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
x' ∉ free_evars (evar_quantify x' n' φ1)
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
x' ∉ free_evars (evar_quantify x' n' φ2) apply IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ1)
IHφ2: ∀ (x : evar) (n : db_index),
  x ∉ free_evars (evar_quantify x n φ2)
x': evar
n': db_index
x' ∉ free_evars (evar_quantify x' n' φ2) apply IHφ2.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), x ∉ free_evars (evar_quantify x n φ)
x': evar
n': db_index
x' ∉ free_evars (evar_quantify x' (S n') φ) apply IHφ.
*
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index), x ∉ free_evars (evar_quantify x n φ)
x': evar
n': db_index
x' ∉ free_evars (evar_quantify x' n' φ) apply IHφ.
Qed.

Theorem svar_quantify_not_free :
forall φ X n, X ∉ (free_svars (svar_quantify X n φ)).
Σ: Signature
∀ (φ : Pattern) (X : svar) (n : db_index),
  X ∉ free_svars (svar_quantify X n φ)
Proof.
Σ: Signature
∀ (φ : Pattern) (X : svar) (n : db_index),
  X ∉ free_svars (svar_quantify X n φ)
induction φ; intros x' n'; simpl; try set_solver.
Σ: Signature
x, x': svar
n': db_index
x'
∉ free_svars
    (if decide (x' = x)
     then patt_bound_svar n'
     else patt_free_svar x)
case_match; simpl; set_solver.
Qed.

(* FIXME: rename! *)
Lemma evar_quantify_noop :
forall φ x n, count_evar_occurrences x φ = 0 ->
evar_quantify x n φ = φ.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  count_evar_occurrences x φ = 0
  → evar_quantify x n φ = φ
Proof.
Σ: Signature
∀ (φ : Pattern) (x : evar) (n : db_index),
  count_evar_occurrences x φ = 0
  → evar_quantify x n φ = φ
induction φ; intros x' n' H; simpl; auto.
Σ: Signature
x, x': evar
n': db_index
H: count_evar_occurrences x' (patt_free_evar x) = 0
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' (patt_app φ1 φ2) = 0
patt_app (evar_quantify x' n' φ1)
  (evar_quantify x' n' φ2) = patt_app φ1 φ2
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' (patt_imp φ1 φ2) = 0
patt_imp (evar_quantify x' n' φ1)
  (evar_quantify x' n' φ2) = 
patt_imp φ1 φ2
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ = 0
  → evar_quantify x n φ = φ
x': evar
n': db_index
H: count_evar_occurrences x' (patt_exists φ) = 0
patt_exists (evar_quantify x' (S n') φ) =
patt_exists φ
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ = 0
  → evar_quantify x n φ = φ
x': evar
n': db_index
H: count_evar_occurrences x' (patt_mu φ) = 0
patt_mu (evar_quantify x' n' φ) = patt_mu φ
-
Σ: Signature
x, x': evar
n': db_index
H: count_evar_occurrences x' (patt_free_evar x) = 0
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x simpl in H.
Σ: Signature
x, x': evar
n': db_index
H: (if decide (x = x') then 1 else 0) = 0
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x
  destruct (decide (x = x')).
Σ: Signature
x, x': evar
n': db_index
e: x = x'
H: 1 = 0
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x
Σ: Signature
x, x': evar
n': db_index
n: x ≠ x'
H: 0 = 0
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x
  +
Σ: Signature
x, x': evar
n': db_index
e: x = x'
H: 1 = 0
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x subst x'.
Σ: Signature
x: evar
n': db_index
H: 1 = 0
(if decide (x = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x destruct (decide (x = x)).
Σ: Signature
x: evar
n': db_index
H: 1 = 0
e: x = x
patt_bound_evar n' = patt_free_evar x
Σ: Signature
x: evar
n': db_index
H: 1 = 0
n: x ≠ x
patt_free_evar x = patt_free_evar x simpl in H.
Σ: Signature
x: evar
n': db_index
H: 1 = 0
e: x = x
patt_bound_evar n' = patt_free_evar x
Σ: Signature
x: evar
n': db_index
H: 1 = 0
n: x ≠ x
patt_free_evar x = patt_free_evar x inversion H.
Σ: Signature
x: evar
n': db_index
H: 1 = 0
n: x ≠ x
patt_free_evar x = patt_free_evar x contradiction.
  +
Σ: Signature
x, x': evar
n': db_index
n: x ≠ x'
H: 0 = 0
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x simpl in H.
Σ: Signature
x, x': evar
n': db_index
n: x ≠ x'
H: 0 = 0
(if decide (x' = x)
 then patt_bound_evar n'
 else patt_free_evar x) = patt_free_evar x destruct (decide (x' = x)); cbn; auto.
Σ: Signature
x, x': evar
n': db_index
n: x ≠ x'
H: 0 = 0
e: x' = x
patt_bound_evar n' = patt_free_evar x congruence.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' (patt_app φ1 φ2) = 0
patt_app (evar_quantify x' n' φ1)
  (evar_quantify x' n' φ2) = patt_app φ1 φ2 simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
patt_app (evar_quantify x' n' φ1)
  (evar_quantify x' n' φ2) = patt_app φ1 φ2 rewrite IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
count_evar_occurrences x' φ1 = 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
patt_app φ1 (evar_quantify x' n' φ2) = patt_app φ1 φ2 lia.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
patt_app φ1 (evar_quantify x' n' φ2) = patt_app φ1 φ2 rewrite IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
count_evar_occurrences x' φ2 = 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
patt_app φ1 φ2 = patt_app φ1 φ2 lia.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
patt_app φ1 φ2 = patt_app φ1 φ2 reflexivity.
-
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' (patt_imp φ1 φ2) = 0
patt_imp (evar_quantify x' n' φ1)
  (evar_quantify x' n' φ2) = patt_imp φ1 φ2 simpl in H.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
patt_imp (evar_quantify x' n' φ1)
  (evar_quantify x' n' φ2) = patt_imp φ1 φ2 rewrite IHφ1.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
count_evar_occurrences x' φ1 = 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
patt_imp φ1 (evar_quantify x' n' φ2) = patt_imp φ1 φ2 lia.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
patt_imp φ1 (evar_quantify x' n' φ2) = patt_imp φ1 φ2 rewrite IHφ2.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
count_evar_occurrences x' φ2 = 0
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
patt_imp φ1 φ2 = patt_imp φ1 φ2 lia.
Σ: Signature
φ1, φ2: Pattern
IHφ1: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ1 = 0
  → evar_quantify x n φ1 = φ1
IHφ2: ∀ (x : evar) (n : db_index),
  count_evar_occurrences x φ2 = 0
  → evar_quantify x n φ2 = φ2
x': evar
n': db_index
H: count_evar_occurrences x' φ1 +
count_evar_occurrences x' φ2 = 0
patt_imp φ1 φ2 = patt_imp φ1 φ2 reflexivity.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
count_evar_occurrences x φ = 0 → evar_quantify x n φ = φ
x': evar
n': db_index
H: count_evar_occurrences x' (patt_exists φ) = 0
patt_exists (evar_quantify x' (S n') φ) =
patt_exists φ simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
count_evar_occurrences x φ = 0 → evar_quantify x n φ = φ
x': evar
n': db_index
H: count_evar_occurrences x' φ = 0
patt_exists (evar_quantify x' (S n') φ) =
patt_exists φ rewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
count_evar_occurrences x φ = 0 → evar_quantify x n φ = φ
x': evar
n': db_index
H: count_evar_occurrences x' φ = 0
count_evar_occurrences x' φ = 0
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
count_evar_occurrences x φ = 0 → evar_quantify x n φ = φ
x': evar
n': db_index
H: count_evar_occurrences x' φ = 0
patt_exists φ = patt_exists φ lia.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
count_evar_occurrences x φ = 0 → evar_quantify x n φ = φ
x': evar
n': db_index
H: count_evar_occurrences x' φ = 0
patt_exists φ = patt_exists φ reflexivity.
-
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
count_evar_occurrences x φ = 0 → evar_quantify x n φ = φ
x': evar
n': db_index
H: count_evar_occurrences x' (patt_mu φ) = 0
patt_mu (evar_quantify x' n' φ) = patt_mu φ simpl in H.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
count_evar_occurrences x φ = 0 → evar_quantify x n φ = φ
x': evar
n': db_index
H: count_evar_occurrences x' φ = 0
patt_mu (evar_quantify x' n' φ) = patt_mu φ rewrite IHφ.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
count_evar_occurrences x φ = 0 → evar_quantify x n φ = φ
x': evar
n': db_index
H: count_evar_occurrences x' φ = 0
count_evar_occurrences x' φ = 0
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
count_evar_occurrences x φ = 0 → evar_quantify x n φ = φ
x': evar
n': db_index
H: count_evar_occurrences x' φ = 0
patt_mu φ = patt_mu φ lia.
Σ: Signature
φ: Pattern
IHφ: ∀ (x : evar) (n : db_index),
count_evar_occurrences x φ = 0 → evar_quantify x n φ = φ
x': evar
n': db_index
H: count_evar_occurrences x' φ = 0
patt_mu φ = patt_mu φ reflexivity.
Qed.


Lemma wf_ex_evar_quantify x p:
well_formed p = true ->
well_formed (patt_exists (evar_quantify x 0 p)) = true.
Σ: Signature
x: evar
p: Pattern
well_formed p = true
→ well_formed (patt_exists (evar_quantify x 0 p)) =
  true
Proof.
Σ: Signature
x: evar
p: Pattern
well_formed p = true
→ well_formed (patt_exists (evar_quantify x 0 p)) =
  true
intros Hwf.
Σ: Signature
x: evar
p: Pattern
Hwf: well_formed p = true
well_formed (patt_exists (evar_quantify x 0 p)) = true
unfold well_formed,well_formed_closed in Hwf.
Σ: Signature
x: evar
p: Pattern
Hwf: [&& well_formed_positive p, well_formed_closed_mu_aux p 0
 & well_formed_closed_ex_aux p 0] = true
well_formed (patt_exists (evar_quantify x 0 p)) = true simpl in Hwf.
Σ: Signature
x: evar
p: Pattern
Hwf: [&& well_formed_positive p, well_formed_closed_mu_aux p 0
 & well_formed_closed_ex_aux p 0] = true
well_formed (patt_exists (evar_quantify x 0 p)) = true
apply andb_prop in Hwf.
Σ: Signature
x: evar
p: Pattern
Hwf: well_formed_positive p = true
∧ well_formed_closed_mu_aux p 0 && well_formed_closed_ex_aux p 0 = true
well_formed (patt_exists (evar_quantify x 0 p)) = true
destruct Hwf as [Hwfp Hwfc].
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed (patt_exists (evar_quantify x 0 p)) = true
simpl in Hwfp.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed (patt_exists (evar_quantify x 0 p)) = true
unfold well_formed,well_formed_closed.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
[&& well_formed_positive
      (patt_exists (evar_quantify x 0 p)),
    well_formed_closed_mu_aux
      (patt_exists (evar_quantify x 0 p)) 0
  & well_formed_closed_ex_aux
      (patt_exists (evar_quantify x 0 p)) 0] = true simpl.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
[&& well_formed_positive (evar_quantify x 0 p),
    well_formed_closed_mu_aux (evar_quantify x 0 p) 0
  & well_formed_closed_ex_aux (evar_quantify x 0 p) 1] =
true
apply andb_true_intro.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_positive (evar_quantify x 0 p) = true
∧ well_formed_closed_mu_aux (evar_quantify x 0 p) 0 &&
  well_formed_closed_ex_aux (evar_quantify x 0 p) 1 =
  true
split.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_positive (evar_quantify x 0 p) = true
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux (evar_quantify x 0 p) 0 &&
well_formed_closed_ex_aux (evar_quantify x 0 p) 1 =
true
-
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_positive (evar_quantify x 0 p) = true simpl.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_positive (evar_quantify x 0 p) = true apply evar_quantify_positive.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_positive p apply Hwfp.
-
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux (evar_quantify x 0 p) 0 &&
well_formed_closed_ex_aux (evar_quantify x 0 p) 1 =
true unfold well_formed_closed.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux (evar_quantify x 0 p) 0 &&
well_formed_closed_ex_aux (evar_quantify x 0 p) 1 =
true
  simpl.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
Hwfc: well_formed_closed_mu_aux p 0 &&
well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux (evar_quantify x 0 p) 0 &&
well_formed_closed_ex_aux (evar_quantify x 0 p) 1 =
true
  destruct_and!.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux (evar_quantify x 0 p) 0 &&
well_formed_closed_ex_aux (evar_quantify x 0 p) 1 =
true
  split_and!.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux (evar_quantify x 0 p) 0 =
true
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_ex_aux (evar_quantify x 0 p) 1 =
true
  +
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux (evar_quantify x 0 p) 0 =
true apply evar_quantify_closed_mu.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_mu_aux p 0 assumption.
  +
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_ex_aux (evar_quantify x 0 p) 1 =
true apply evar_quantify_closed_ex.
Σ: Signature
x: evar
p: Pattern
Hwfp: well_formed_positive p = true
H: well_formed_closed_mu_aux p 0 = true
H0: well_formed_closed_ex_aux p 0 = true
well_formed_closed_ex_aux p 0 assumption.
Qed.


Lemma free_evars_evar_quantify x n p:
  free_evars (evar_quantify x n p) = free_evars p ∖ {[x]}.
Σ: Signature
x: evar
n: db_index
p: Pattern
free_evars (evar_quantify x n p) =
free_evars p ∖ {[x]}
Proof.
Σ: Signature
x: evar
n: db_index
p: Pattern
free_evars (evar_quantify x n p) =
free_evars p ∖ {[x]}
  move: n.
Σ: Signature
x: evar
p: Pattern
∀ n : db_index,
  free_evars (evar_quantify x n p) =
  free_evars p ∖ {[x]}
  induction p; intros n'; simpl; try set_solver.
Σ: Signature
x, x0: evar
n': db_index
free_evars
  (if decide (x = x0)
   then patt_bound_evar n'
   else patt_free_evar x0) = {[x0]} ∖ {[x]}
  destruct (decide (x = x0)).
Σ: Signature
x, x0: evar
n': db_index
e: x = x0
free_evars (patt_bound_evar n') = {[x0]} ∖ {[x]}
Σ: Signature
x, x0: evar
n': db_index
n: x ≠ x0
free_evars (patt_free_evar x0) = {[x0]} ∖ {[x]}
    +
Σ: Signature
x, x0: evar
n': db_index
e: x = x0
free_evars (patt_bound_evar n') = {[x0]} ∖ {[x]} subst.
Σ: Signature
x0: evar
n': db_index
free_evars (patt_bound_evar n') = {[x0]} ∖ {[x0]} simpl.
Σ: Signature
x0: evar
n': db_index
∅ = {[x0]} ∖ {[x0]} set_solver.
    +
Σ: Signature
x, x0: evar
n': db_index
n: x ≠ x0
free_evars (patt_free_evar x0) = {[x0]} ∖ {[x]} simpl.
Σ: Signature
x, x0: evar
n': db_index
n: x ≠ x0
{[x0]} = {[x0]} ∖ {[x]} set_solver.
Qed.

Lemma free_svars_svar_quantify X n p:
  free_svars (svar_quantify X n p) = free_svars p ∖ {[X]}.
Σ: Signature
X: svar
n: db_index
p: Pattern
free_svars (svar_quantify X n p) =
free_svars p ∖ {[X]}
Proof.
Σ: Signature
X: svar
n: db_index
p: Pattern
free_svars (svar_quantify X n p) =
free_svars p ∖ {[X]}
  move: n.
Σ: Signature
X: svar
p: Pattern
∀ n : db_index,
  free_svars (svar_quantify X n p) =
  free_svars p ∖ {[X]}
  induction p; intros n'; simpl; try set_solver.
Σ: Signature
X, x: svar
n': db_index
free_svars
  (if decide (X = x)
   then patt_bound_svar n'
   else patt_free_svar x) = {[x]} ∖ {[X]}
  destruct (decide (X = x)).
Σ: Signature
X, x: svar
n': db_index
e: X = x
free_svars (patt_bound_svar n') = {[x]} ∖ {[X]}
Σ: Signature
X, x: svar
n': db_index
n: X ≠ x
free_svars (patt_free_svar x) = {[x]} ∖ {[X]}
    +
Σ: Signature
X, x: svar
n': db_index
e: X = x
free_svars (patt_bound_svar n') = {[x]} ∖ {[X]} subst.
Σ: Signature
x: svar
n': db_index
free_svars (patt_bound_svar n') = {[x]} ∖ {[x]} simpl.
Σ: Signature
x: svar
n': db_index
∅ = {[x]} ∖ {[x]} set_solver.
    +
Σ: Signature
X, x: svar
n': db_index
n: X ≠ x
free_svars (patt_free_svar x) = {[x]} ∖ {[X]} simpl.
Σ: Signature
X, x: svar
n': db_index
n: X ≠ x
{[x]} = {[x]} ∖ {[X]} set_solver.
Qed.


Lemma no_neg_occ_db_bsvar_subst phi psi dbi1 dbi2:
well_formed_closed_mu_aux psi 0 = true -> dbi1 < dbi2 ->
(no_negative_occurrence_db_b dbi1 phi = true ->
 no_negative_occurrence_db_b dbi1 (bsvar_subst phi psi dbi2) = true)
/\ (no_positive_occurrence_db_b dbi1 phi = true ->
    no_positive_occurrence_db_b dbi1 (bsvar_subst phi psi dbi2) = true).
Σ: Signature
phi, psi: Pattern
dbi1, dbi2: nat
well_formed_closed_mu_aux psi 0 = true
→ dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi = true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi = true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
Proof.
Σ: Signature
phi, psi: Pattern
dbi1, dbi2: nat
well_formed_closed_mu_aux psi 0 = true
→ dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi = true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi = true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
intros Hwfcpsi.
Σ: Signature
phi, psi: Pattern
dbi1, dbi2: nat
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi = true
   → no_negative_occurrence_db_b dbi1
       phi^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi = true
     → no_positive_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
move: dbi1 dbi2.
Σ: Signature
phi, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi = true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi = true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)

induction phi; intros dbi1 dbi2 H; simpl; auto.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_bound_svar n) =
 true
 → no_negative_occurrence_db_b dbi1
     match compare_nat n dbi2 with
     | Nat_less _ _ _ => patt_bound_svar n
     | Nat_equal _ _ _ => psi
     | Nat_greater _ _ _ =>
         patt_bound_svar (Nat.pred n)
     end = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_bound_svar n) = true
   → no_positive_occurrence_db_b dbi1
       match compare_nat n dbi2 with
       | Nat_less _ _ _ => patt_bound_svar n
       | Nat_equal _ _ _ => psi
       | Nat_greater _ _ _ =>
           patt_bound_svar (Nat.pred n)
       end = true)
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi1: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi1 =
     true
     → no_negative_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi1 =
       true
       → no_positive_occurrence_db_b dbi1
           phi1^[svar:dbi2↦psi] = true)
IHphi2: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi2 =
     true
     → no_negative_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi2 =
       true
       → no_positive_occurrence_db_b dbi1
           phi2^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_app phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_app phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_app phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi1: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi1 =
     true
     → no_negative_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi1 =
       true
       → no_positive_occurrence_db_b dbi1
           phi1^[svar:dbi2↦psi] = true)
IHphi2: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi2 =
     true
     → no_negative_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi2 =
       true
       → no_positive_occurrence_db_b dbi1
           phi2^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_imp phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_imp phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_imp phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
Σ: Signature
phi, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi =
     true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi =
       true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_exists phi) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_exists phi^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1 (patt_exists phi) =
   true
   → no_positive_occurrence_db_b dbi1
       (patt_exists phi^[svar:dbi2↦psi]) = true)
Σ: Signature
phi, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi =
     true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi =
       true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_mu phi) = true
 → no_negative_occurrence_db_b dbi1
     (patt_mu phi^[svar:S dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1 (patt_mu phi) =
   true
   → no_positive_occurrence_db_b dbi1
       (patt_mu phi^[svar:S dbi2↦psi]) = true)
-
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_bound_svar n) =
 true
 → no_negative_occurrence_db_b dbi1
     match compare_nat n dbi2 with
     | Nat_less _ _ _ => patt_bound_svar n
     | Nat_equal _ _ _ => psi
     | Nat_greater _ _ _ =>
         patt_bound_svar (Nat.pred n)
     end = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_bound_svar n) = true
   → no_positive_occurrence_db_b dbi1
       match compare_nat n dbi2 with
       | Nat_less _ _ _ => patt_bound_svar n
       | Nat_equal _ _ _ => psi
       | Nat_greater _ _ _ =>
           patt_bound_svar (Nat.pred n)
       end = true)
  case_match; auto.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
(no_negative_occurrence_db_b dbi1 (patt_bound_svar n) =
 true → no_negative_occurrence_db_b dbi1 psi = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_bound_svar n) = true
   → no_positive_occurrence_db_b dbi1 psi = true)
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
(no_negative_occurrence_db_b dbi1 (patt_bound_svar n) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_bound_svar (Nat.pred n)) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_bound_svar n) = true
   → no_positive_occurrence_db_b dbi1
       (patt_bound_svar (Nat.pred n)) = true)
  +
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
(no_negative_occurrence_db_b dbi1 (patt_bound_svar n) =
 true → no_negative_occurrence_db_b dbi1 psi = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_bound_svar n) = true
   → no_positive_occurrence_db_b dbi1 psi = true) split; intros H0'.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
H0': no_negative_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_negative_occurrence_db_b dbi1 psi = true
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_positive_occurrence_db_b dbi1 psi = true
    *
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
H0': no_negative_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_negative_occurrence_db_b dbi1 psi = true apply wfc_impl_no_neg_occ.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
H0': no_negative_occurrence_db_b dbi1
  (patt_bound_svar n) = true
well_formed_closed_mu_aux psi 0 = true apply Hwfcpsi.
    *
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_positive_occurrence_db_b dbi1 psi = true apply wfc_impl_no_pos_occ.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
e: n = dbi2
H0: compare_nat n dbi2 = Nat_equal n dbi2 e
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
well_formed_closed_mu_aux psi 0 = true apply Hwfcpsi.
  +
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
(no_negative_occurrence_db_b dbi1 (patt_bound_svar n) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_bound_svar (Nat.pred n)) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_bound_svar n) = true
   → no_positive_occurrence_db_b dbi1
       (patt_bound_svar (Nat.pred n)) = true) split; intros H0'.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_negative_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_negative_occurrence_db_b dbi1
  (patt_bound_svar (Nat.pred n)) = true
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_positive_occurrence_db_b dbi1
  (patt_bound_svar (Nat.pred n)) = true
    *
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_negative_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_negative_occurrence_db_b dbi1
  (patt_bound_svar (Nat.pred n)) = true auto.
    *
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
no_positive_occurrence_db_b dbi1
  (patt_bound_svar (Nat.pred n)) = true cbn.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
(if decide (Nat.pred n = dbi1) then false else true) =
true repeat case_match.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
e: Nat.pred n = dbi1
H1: decide (Nat.pred n = dbi1) = left e
false = true
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
n0: Nat.pred n ≠ dbi1
H1: decide (Nat.pred n = dbi1) = right n0
true = true lia.
Σ: Signature
n: db_index
psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
H: dbi1 < dbi2
g: n > dbi2
H0: compare_nat n dbi2 = Nat_greater n dbi2 g
H0': no_positive_occurrence_db_b dbi1
  (patt_bound_svar n) = true
n0: Nat.pred n ≠ dbi1
H1: decide (Nat.pred n = dbi1) = right n0
true = true reflexivity.
-
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi1: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi1 =
     true
     → no_negative_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi1 =
       true
       → no_positive_occurrence_db_b dbi1
           phi1^[svar:dbi2↦psi] = true)
IHphi2: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi2 =
     true
     → no_negative_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi2 =
       true
       → no_positive_occurrence_db_b dbi1
           phi2^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_app phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_app phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_app phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true) specialize (IHphi1 dbi1 dbi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi2 =
     true
     → no_negative_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi2 =
       true
       → no_positive_occurrence_db_b dbi1
           phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_app phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_app phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_app phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  specialize (IHphi2 dbi1 dbi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_app phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_app phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_app phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  destruct (IHphi1 H) as [IHphi11 IHphi12].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
(no_negative_occurrence_db_b dbi1 (patt_app phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_app phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_app phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  destruct (IHphi2 H) as [IHphi21 IHphi22].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
(no_negative_occurrence_db_b dbi1 (patt_app phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_app phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_app phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  split; intro H0.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_negative_occurrence_db_b dbi1
  (patt_app phi1 phi2) = true
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_positive_occurrence_db_b dbi1
  (patt_app phi1 phi2) = true
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_negative_occurrence_db_b dbi1
  (patt_app phi1 phi2) = true
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true eapply elimT in H0.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: ?P
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_negative_occurrence_db_b dbi1
  (patt_app phi1 phi2) = true
ssrbool.reflect ?P
  (no_negative_occurrence_db_b dbi1
     (patt_app phi1 phi2))
    2: apply andP.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
∧ (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi) then false else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    destruct H0 as [H1 H2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi11 H1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi21 H2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    cbn.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1 phi1^[svar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[svar:dbi2↦psi] =
true
    rewrite IHphi11 IHphi21.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
true && true = true reflexivity.
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_positive_occurrence_db_b dbi1
  (patt_app phi1 phi2) = true
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true eapply elimT in H0.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: ?P
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_positive_occurrence_db_b dbi1
  (patt_app phi1 phi2) = true
ssrbool.reflect ?P
  (no_positive_occurrence_db_b dbi1
     (patt_app phi1 phi2))
    2: apply andP.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
∧ (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi) then false else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    destruct H0 as [H1 H2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi12 H1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi22 H2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_app phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    cbn.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1 phi1^[svar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[svar:dbi2↦psi] =
true
    rewrite IHphi12 IHphi22.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
true && true = true reflexivity.
-
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi1: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi1 =
     true
     → no_negative_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi1 =
       true
       → no_positive_occurrence_db_b dbi1
           phi1^[svar:dbi2↦psi] = true)
IHphi2: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi2 =
     true
     → no_negative_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi2 =
       true
       → no_positive_occurrence_db_b dbi1
           phi2^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_imp phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_imp phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_imp phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true) specialize (IHphi1 dbi1 dbi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi2 =
     true
     → no_negative_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi2 =
       true
       → no_positive_occurrence_db_b dbi1
           phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_imp phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_imp phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_imp phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  specialize (IHphi2 dbi1 dbi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_imp phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_imp phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_imp phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  destruct (IHphi1 H) as [IHphi11 IHphi12].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
(no_negative_occurrence_db_b dbi1 (patt_imp phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_imp phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_imp phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  destruct (IHphi2 H) as [IHphi21 IHphi22].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
(no_negative_occurrence_db_b dbi1 (patt_imp phi1 phi2) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_imp phi1^[svar:dbi2↦psi]
        phi2^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2) = true
   → no_positive_occurrence_db_b dbi1
       (patt_imp phi1^[svar:dbi2↦psi]
          phi2^[svar:dbi2↦psi]) = true)
  split; intro H0.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_negative_occurrence_db_b dbi1
  (patt_imp phi1 phi2) = true
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_positive_occurrence_db_b dbi1
  (patt_imp phi1 phi2) = true
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_negative_occurrence_db_b dbi1
  (patt_imp phi1 phi2) = true
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true eapply elimT in H0.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: ?P
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_negative_occurrence_db_b dbi1
  (patt_imp phi1 phi2) = true
ssrbool.reflect ?P
  (no_negative_occurrence_db_b dbi1
     (patt_imp phi1 phi2))
    2: apply andP.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
∧ (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi) then false else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    destruct H0 as [H1 H2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi12 H1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi21 H2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    cbn.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[svar:dbi2↦psi] =
true fold no_negative_occurrence_db_b no_positive_occurrence_db_b.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1 phi1^[svar:dbi2↦psi] &&
no_negative_occurrence_db_b dbi1 phi2^[svar:dbi2↦psi] =
true
    rewrite IHphi12 IHphi21.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi2
true && true = true reflexivity.
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_positive_occurrence_db_b dbi1
  (patt_imp phi1 phi2) = true
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true eapply elimT in H0.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: ?P
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: no_positive_occurrence_db_b dbi1
  (patt_imp phi1 phi2) = true
ssrbool.reflect ?P
  (no_positive_occurrence_db_b dbi1
     (patt_imp phi1 phi2))
    2: apply andP.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H0: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
∧ (fix no_negative_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_app ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_negative_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_negative_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_negative_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   with no_positive_occurrence_db_b
     (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
       bool :=
     match ϕ with
     | patt_bound_svar n =>
         if decide (n = dbi) then false else true
     | patt_app ϕ₁ ϕ₂ =>
         no_positive_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_imp ϕ₁ ϕ₂ =>
         no_negative_occurrence_db_b dbi ϕ₁ &&
         no_positive_occurrence_db_b dbi ϕ₂
     | patt_exists ϕ' =>
         no_positive_occurrence_db_b dbi ϕ'
     | patt_mu ϕ' =>
         no_positive_occurrence_db_b (S dbi) ϕ'
     | _ => true
     end
   for
   no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    destruct H0 as [H1 H2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1 phi1 = true
→ no_negative_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi11 H1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1 phi2 = true
→ no_positive_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    specialize (IHphi22 H2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_positive_occurrence_db_b dbi1
  (patt_imp phi1^[svar:dbi2↦psi] phi2^[svar:dbi2↦psi]) =
true
    cbn.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1
  phi1^[svar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[svar:dbi2↦psi] =
true fold no_negative_occurrence_db_b no_positive_occurrence_db_b.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
no_negative_occurrence_db_b dbi1 phi1^[svar:dbi2↦psi] &&
no_positive_occurrence_db_b dbi1 phi2^[svar:dbi2↦psi] =
true
    rewrite IHphi11 IHphi22.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
dbi1, dbi2: nat
IHphi1: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi1 =
   true
   → no_negative_occurrence_db_b dbi1
       phi1^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi1 =
     true
     → no_positive_occurrence_db_b dbi1
         phi1^[svar:dbi2↦psi] = true)
IHphi2: dbi1 < dbi2
→ (no_negative_occurrence_db_b dbi1 phi2 =
   true
   → no_negative_occurrence_db_b dbi1
       phi2^[svar:dbi2↦psi] = true)
  ∧ (no_positive_occurrence_db_b dbi1 phi2 =
     true
     → no_positive_occurrence_db_b dbi1
         phi2^[svar:dbi2↦psi] = true)
H: dbi1 < dbi2
IHphi11: no_negative_occurrence_db_b dbi1
  phi1^[svar:dbi2↦psi] = true
IHphi12: no_positive_occurrence_db_b dbi1 phi1 = true
→ no_positive_occurrence_db_b dbi1
    phi1^[svar:dbi2↦psi] = true
IHphi21: no_negative_occurrence_db_b dbi1 phi2 = true
→ no_negative_occurrence_db_b dbi1
    phi2^[svar:dbi2↦psi] = true
IHphi22: no_positive_occurrence_db_b dbi1
  phi2^[svar:dbi2↦psi] = true
H1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) dbi1 phi1
H2: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) dbi1 phi2
true && true = true reflexivity.
-
Σ: Signature
phi, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi =
     true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi =
       true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_exists phi) =
 true
 → no_negative_occurrence_db_b dbi1
     (patt_exists phi^[svar:dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1 (patt_exists phi) =
   true
   → no_positive_occurrence_db_b dbi1
       (patt_exists phi^[svar:dbi2↦psi]) = true) split; intros H0; apply IHphi; auto; lia.
-
Σ: Signature
phi, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi =
     true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi =
       true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
(no_negative_occurrence_db_b dbi1 (patt_mu phi) = true
 → no_negative_occurrence_db_b dbi1
     (patt_mu phi^[svar:S dbi2↦psi]) = true)
∧ (no_positive_occurrence_db_b dbi1 (patt_mu phi) =
   true
   → no_positive_occurrence_db_b dbi1
       (patt_mu phi^[svar:S dbi2↦psi]) = true) apply IHphi.
Σ: Signature
phi, psi: Pattern
Hwfcpsi: well_formed_closed_mu_aux psi 0 = true
IHphi: ∀ dbi1 dbi2 : nat,
  dbi1 < dbi2
  → (no_negative_occurrence_db_b dbi1 phi =
     true
     → no_negative_occurrence_db_b dbi1
         phi^[svar:dbi2↦psi] = true)
    ∧ (no_positive_occurrence_db_b dbi1 phi =
       true
       → no_positive_occurrence_db_b dbi1
           phi^[svar:dbi2↦psi] = true)
dbi1, dbi2: nat
H: dbi1 < dbi2
S dbi1 < S dbi2 lia.
Qed.


Lemma Private_wfp_bsvar_subst (phi psi : Pattern) (n : nat) :
well_formed_positive psi ->
well_formed_closed_mu_aux psi 0 ->
well_formed_positive phi ->
(
  no_negative_occurrence_db_b n phi ->
  well_formed_positive (bsvar_subst phi psi n) )
/\ (no_positive_occurrence_db_b n phi ->
    forall phi',
      well_formed_positive phi' ->
      well_formed_positive (patt_imp (bsvar_subst phi psi n) phi')
   )
.
Σ: Signature
phi, psi: Pattern
n: nat
well_formed_positive psi
→ well_formed_closed_mu_aux psi 0
  → well_formed_positive phi
    → (no_negative_occurrence_db_b n phi
       → well_formed_positive phi^[svar:n↦psi])
      ∧ (no_positive_occurrence_db_b n phi
         → ∀ phi' : Pattern,
             well_formed_positive phi'
             → well_formed_positive
                 (patt_imp phi^[svar:n↦psi] phi'))
Proof.
Σ: Signature
phi, psi: Pattern
n: nat
well_formed_positive psi
→ well_formed_closed_mu_aux psi 0
  → well_formed_positive phi
    → (no_negative_occurrence_db_b n phi
       → well_formed_positive phi^[svar:n↦psi])
      ∧ (no_positive_occurrence_db_b n phi
         → ∀ phi' : Pattern,
             well_formed_positive phi'
             → well_formed_positive
                 (patt_imp phi^[svar:n↦psi] phi'))
intros Hwfppsi Hwfcpsi.
Σ: Signature
phi, psi: Pattern
n: nat
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
well_formed_positive phi
→ (no_negative_occurrence_db_b n phi
   → well_formed_positive phi^[svar:n↦psi])
  ∧ (no_positive_occurrence_db_b n phi
     → ∀ phi' : Pattern,
         well_formed_positive phi'
         → well_formed_positive
             (patt_imp phi^[svar:n↦psi] phi'))
move: n.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               (patt_imp phi^[svar:n↦psi] phi'))
induction phi; intros n' Hwfpphi; cbn in *; auto.
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
(true
 → well_formed_positive
     match compare_nat n n' with
     | Nat_less _ _ _ => patt_bound_svar n
     | Nat_equal _ _ _ => psi
     | Nat_greater _ _ _ =>
         patt_bound_svar (Nat.pred n)
     end)
∧ ((if decide (n = n') then false else true)
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           match compare_nat n n' with
           | Nat_less _ _ _ => patt_bound_svar n
           | Nat_equal _ _ _ => psi
           | Nat_greater _ _ _ =>
               patt_bound_svar (Nat.pred n)
           end && well_formed_positive phi')
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
(no_negative_occurrence_db_b n' phi1 &&
 no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi1^[svar:n'↦psi] &&
   well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1 &&
   no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive phi1^[svar:n'↦psi] &&
         well_formed_positive phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
((fix no_negative_occurrence_db_b
    (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
      bool :=
    match ϕ with
    | patt_app ϕ₁ ϕ₂ =>
        no_negative_occurrence_db_b dbi ϕ₁ &&
        no_negative_occurrence_db_b dbi ϕ₂
    | patt_imp ϕ₁ ϕ₂ =>
        no_positive_occurrence_db_b dbi ϕ₁ &&
        no_negative_occurrence_db_b dbi ϕ₂
    | patt_exists ϕ' =>
        no_negative_occurrence_db_b dbi ϕ'
    | patt_mu ϕ' =>
        no_negative_occurrence_db_b (S dbi) ϕ'
    | _ => true
    end
  with no_positive_occurrence_db_b
    (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
      bool :=
    match ϕ with
    | patt_bound_svar n =>
        if decide (n = dbi) then false else true
    | patt_app ϕ₁ ϕ₂ =>
        no_positive_occurrence_db_b dbi ϕ₁ &&
        no_positive_occurrence_db_b dbi ϕ₂
    | patt_imp ϕ₁ ϕ₂ =>
        no_negative_occurrence_db_b dbi ϕ₁ &&
        no_positive_occurrence_db_b dbi ϕ₂
    | patt_exists ϕ' =>
        no_positive_occurrence_db_b dbi ϕ'
    | patt_mu ϕ' =>
        no_positive_occurrence_db_b (S dbi) ϕ'
    | _ => true
    end
  for
  no_positive_occurrence_db_b) n' phi1 &&
 no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi1^[svar:n'↦psi] &&
   well_formed_positive phi2^[svar:n'↦psi])
∧ ((fix no_negative_occurrence_db_b
      (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
        bool :=
      match ϕ with
      | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
      | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
      | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
      | patt_mu ϕ' =>
          no_negative_occurrence_db_b (S dbi) ϕ'
      | _ => true
      end
    with no_positive_occurrence_db_b
      (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
        bool :=
      match ϕ with
      | patt_bound_svar n =>
          if decide (n = dbi) then false else true
      | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
      | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
      | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
      | patt_mu ϕ' =>
          no_positive_occurrence_db_b (S dbi) ϕ'
      | _ => true
      end
    for
    no_negative_occurrence_db_b) n' phi1 &&
   no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive phi1^[svar:n'↦psi] &&
         well_formed_positive phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
-
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
(true
 → well_formed_positive
     match compare_nat n n' with
     | Nat_less _ _ _ => patt_bound_svar n
     | Nat_equal _ _ _ => psi
     | Nat_greater _ _ _ =>
         patt_bound_svar (Nat.pred n)
     end)
∧ ((if decide (n = n') then false else true)
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           match compare_nat n n' with
           | Nat_less _ _ _ => patt_bound_svar n
           | Nat_equal _ _ _ => psi
           | Nat_greater _ _ _ =>
               patt_bound_svar (Nat.pred n)
           end && well_formed_positive phi') split.
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
true
→ well_formed_positive
    match compare_nat n n' with
    | Nat_less _ _ _ => patt_bound_svar n
    | Nat_equal _ _ _ => psi
    | Nat_greater _ _ _ =>
        patt_bound_svar (Nat.pred n)
    end
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
(if decide (n = n') then false else true)
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        match compare_nat n n' with
        | Nat_less _ _ _ => patt_bound_svar n
        | Nat_equal _ _ _ => psi
        | Nat_greater _ _ _ =>
            patt_bound_svar (Nat.pred n)
        end && well_formed_positive phi'
  +
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
true
→ well_formed_positive
    match compare_nat n n' with
    | Nat_less _ _ _ => patt_bound_svar n
    | Nat_equal _ _ _ => psi
    | Nat_greater _ _ _ =>
        patt_bound_svar (Nat.pred n)
    end intros _.
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end case_match; auto.
  +
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
(if decide (n = n') then false else true)
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        match compare_nat n n' with
        | Nat_less _ _ _ => patt_bound_svar n
        | Nat_equal _ _ _ => psi
        | Nat_greater _ _ _ =>
            patt_bound_svar (Nat.pred n)
        end && well_formed_positive phi' intros H phi' Hwfphi'.
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
H: if decide (n = n') then false else true
phi': Pattern
Hwfphi': well_formed_positive phi'
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end && well_formed_positive phi'
    cbn in *.
Σ: Signature
n: db_index
psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
Hwfpphi: true
H: if decide (n = n') then false else true
phi': Pattern
Hwfphi': well_formed_positive phi'
well_formed_positive
  match compare_nat n n' with
  | Nat_less _ _ _ => patt_bound_svar n
  | Nat_equal _ _ _ => psi
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
  end && well_formed_positive phi'
    do 2 case_match; auto.
-
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
(no_negative_occurrence_db_b n' phi1 &&
 no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi1^[svar:n'↦psi] &&
   well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1 &&
   no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive phi1^[svar:n'↦psi] &&
         well_formed_positive phi2^[svar:n'↦psi] &&
         well_formed_positive phi') split.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
no_negative_occurrence_db_b n' phi1 &&
no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi1^[svar:n'↦psi] &&
  well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
no_positive_occurrence_db_b n' phi1 &&
no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive phi1^[svar:n'↦psi] &&
      well_formed_positive phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
no_negative_occurrence_db_b n' phi1 &&
no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi1^[svar:n'↦psi] &&
  well_formed_positive phi2^[svar:n'↦psi] intros Hnoneg.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoneg: no_negative_occurrence_db_b n' phi1 &&
no_negative_occurrence_db_b n' phi2
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    apply andb_prop in Hnoneg.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoneg: no_negative_occurrence_db_b n' phi1 = true
∧ no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] destruct Hnoneg as [Hnoneg1 Hnoneg2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    apply andb_prop in Hwfpphi.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 = true
∧ well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] destruct Hwfpphi as [Hwfpphi1 Hwfpphi2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    
    specialize (IHphi1 n' Hwfpphi1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    destruct IHphi1 as [IHphi11 IHphi12].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    specialize (IHphi11 Hnoneg1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    rewrite IHphi11.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true && well_formed_positive phi2^[svar:n'↦psi]

    specialize (IHphi2 n' Hwfpphi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true && well_formed_positive phi2^[svar:n'↦psi]
    destruct IHphi2 as [IHphi21 IHphi22].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true && well_formed_positive phi2^[svar:n'↦psi]
    specialize (IHphi21 Hnoneg2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true && well_formed_positive phi2^[svar:n'↦psi]
    rewrite IHphi21.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnoneg1: no_negative_occurrence_db_b n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true && true
    auto.
    
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
no_positive_occurrence_db_b n' phi1 &&
no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive phi1^[svar:n'↦psi] &&
      well_formed_positive phi2^[svar:n'↦psi] &&
      well_formed_positive phi' intros Hnopos.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnopos: no_positive_occurrence_db_b n' phi1 &&
no_positive_occurrence_db_b n' phi2
∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi'
    apply andb_prop in Hnopos.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnopos: no_positive_occurrence_db_b n' phi1 = true
∧ no_positive_occurrence_db_b n' phi2 = true
∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi' destruct Hnopos as [Hnopos1 Hnopos2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi'
    apply andb_prop in Hwfpphi.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 = true
∧ well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi' destruct Hwfpphi as [Hwfpphi1 Hwfpphi2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi'
    intros phi' Hwfpphi'.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'

    specialize (IHphi1 n' Hwfpphi1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    specialize (IHphi2 n' Hwfpphi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    destruct IHphi1 as [IHphi11 IHphi12].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    destruct IHphi2 as [IHphi21 IHphi22].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    rewrite IHphi12.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
no_positive_occurrence_db_b n' phi1
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi' fold no_negative_occurrence_db_b no_positive_occurrence_db_b in *.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
no_positive_occurrence_db_b n' phi1
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi'
    {
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
no_positive_occurrence_db_b n' phi1 rewrite Hnopos1.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true auto. }
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi'
    specialize (IHphi22 Hnopos2 phi' Hwfpphi').
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
phi': Pattern
IHphi22: well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
Hwfpphi': well_formed_positive phi'
well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi'
    apply andb_prop in IHphi22.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
phi': Pattern
IHphi22: well_formed_positive phi2^[svar:n'↦psi] =
true ∧ well_formed_positive phi' = true
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
Hwfpphi': well_formed_positive phi'
well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi' destruct IHphi22 as [IHphi221 IHphi222].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
phi': Pattern
IHphi221: well_formed_positive phi2^[svar:n'↦psi] =
true
IHphi222: well_formed_positive phi' = true
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
Hwfpphi': well_formed_positive phi'
well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi'
    rewrite IHphi221.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
phi': Pattern
IHphi221: well_formed_positive phi2^[svar:n'↦psi] =
true
IHphi222: well_formed_positive phi' = true
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
Hwfpphi': well_formed_positive phi'
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi' auto.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi'
    rewrite Hwfpphi'.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: no_positive_occurrence_db_b n' phi1 = true
Hnopos2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && true auto.

-
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
((fix no_negative_occurrence_db_b
    (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
      bool :=
    match ϕ with
    | patt_app ϕ₁ ϕ₂ =>
        no_negative_occurrence_db_b dbi ϕ₁ &&
        no_negative_occurrence_db_b dbi ϕ₂
    | patt_imp ϕ₁ ϕ₂ =>
        no_positive_occurrence_db_b dbi ϕ₁ &&
        no_negative_occurrence_db_b dbi ϕ₂
    | patt_exists ϕ' =>
        no_negative_occurrence_db_b dbi ϕ'
    | patt_mu ϕ' =>
        no_negative_occurrence_db_b (S dbi) ϕ'
    | _ => true
    end
  with no_positive_occurrence_db_b
    (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
      bool :=
    match ϕ with
    | patt_bound_svar n =>
        if decide (n = dbi) then false else true
    | patt_app ϕ₁ ϕ₂ =>
        no_positive_occurrence_db_b dbi ϕ₁ &&
        no_positive_occurrence_db_b dbi ϕ₂
    | patt_imp ϕ₁ ϕ₂ =>
        no_negative_occurrence_db_b dbi ϕ₁ &&
        no_positive_occurrence_db_b dbi ϕ₂
    | patt_exists ϕ' =>
        no_positive_occurrence_db_b dbi ϕ'
    | patt_mu ϕ' =>
        no_positive_occurrence_db_b (S dbi) ϕ'
    | _ => true
    end
  for
  no_positive_occurrence_db_b) n' phi1 &&
 no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi1^[svar:n'↦psi] &&
   well_formed_positive phi2^[svar:n'↦psi])
∧ ((fix no_negative_occurrence_db_b
      (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
        bool :=
      match ϕ with
      | patt_app ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
      | patt_imp ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_negative_occurrence_db_b dbi ϕ₂
      | patt_exists ϕ' =>
          no_negative_occurrence_db_b dbi ϕ'
      | patt_mu ϕ' =>
          no_negative_occurrence_db_b (S dbi) ϕ'
      | _ => true
      end
    with no_positive_occurrence_db_b
      (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
        bool :=
      match ϕ with
      | patt_bound_svar n =>
          if decide (n = dbi) then false else true
      | patt_app ϕ₁ ϕ₂ =>
          no_positive_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
      | patt_imp ϕ₁ ϕ₂ =>
          no_negative_occurrence_db_b dbi ϕ₁ &&
          no_positive_occurrence_db_b dbi ϕ₂
      | patt_exists ϕ' =>
          no_positive_occurrence_db_b dbi ϕ'
      | patt_mu ϕ' =>
          no_positive_occurrence_db_b (S dbi) ϕ'
      | _ => true
      end
    for
    no_negative_occurrence_db_b) n' phi1 &&
   no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive phi1^[svar:n'↦psi] &&
         well_formed_positive phi2^[svar:n'↦psi] &&
         well_formed_positive phi') split.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 &&
no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi1^[svar:n'↦psi] &&
  well_formed_positive phi2^[svar:n'↦psi]
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 &&
no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive phi1^[svar:n'↦psi] &&
      well_formed_positive phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 &&
no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi1^[svar:n'↦psi] &&
  well_formed_positive phi2^[svar:n'↦psi] intros Hnoposneg.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoposneg: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 &&
no_negative_occurrence_db_b n' phi2
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    apply andb_prop in Hnoposneg.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoposneg: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 =
true
∧ no_negative_occurrence_db_b n' phi2 =
  true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] destruct Hnoposneg as [Hnopos1 Hnoneg2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    apply andb_prop in Hwfpphi.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 = true
∧ well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] destruct Hwfpphi as [Hwfpphi1 Hwfpphi2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    specialize (IHphi1 n' Hwfpphi1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    specialize (IHphi2 n' Hwfpphi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    destruct IHphi1 as [IHphi11 IHphi12].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    destruct IHphi2 as [IHphi21 IHphi22].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    specialize (IHphi12 Hnopos1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] clear IHphi11.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    specialize (IHphi21 Hnoneg2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi'
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] clear IHphi22.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi'
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi]
    rewrite IHphi21.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi1^[svar:n'↦psi] &&
    well_formed_positive phi'
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] && true

    specialize (IHphi12 patt_bott).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: well_formed_positive patt_bott
→ well_formed_positive phi1^[svar:n'↦psi] &&
  well_formed_positive patt_bott
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] && true simpl in IHphi12.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: true
→ well_formed_positive phi1^[svar:n'↦psi] &&
  true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
well_formed_positive phi1^[svar:n'↦psi] && true
    assert (Htrue: is_true true).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: true
→ well_formed_positive phi1^[svar:n'↦psi] &&
  true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: true
→ well_formed_positive phi1^[svar:n'↦psi] &&
  true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
Htrue: true
well_formed_positive phi1^[svar:n'↦psi] && true
    {
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: true
→ well_formed_positive phi1^[svar:n'↦psi] &&
  true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
true auto. }
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: true
→ well_formed_positive phi1^[svar:n'↦psi] &&
  true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
Htrue: true
well_formed_positive phi1^[svar:n'↦psi] && true
    specialize (IHphi12 Htrue).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: well_formed_positive phi1^[svar:n'↦psi] &&
true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
Htrue: true
well_formed_positive phi1^[svar:n'↦psi] && true
    rewrite IHphi12.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi12: well_formed_positive phi1^[svar:n'↦psi] &&
true
IHphi21: well_formed_positive phi2^[svar:n'↦psi]
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_positive_occurrence_db_b) n' phi1 = true
Hnoneg2: no_negative_occurrence_db_b n' phi2 = true
Htrue: true
true
    auto.
  +
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
(fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi) then false else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 &&
no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive phi1^[svar:n'↦psi] &&
      well_formed_positive phi2^[svar:n'↦psi] &&
      well_formed_positive phi' intros Hnoposneg phi' Hwfpphi'.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoposneg: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 &&
no_positive_occurrence_db_b n' phi2
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    apply andb_prop in Hnoposneg.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnoposneg: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct
    ϕ} : bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 =
true
∧ no_positive_occurrence_db_b n' phi2 =
  true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi' destruct Hnoposneg as [Hnopos1 Hnoneg2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 &&
well_formed_positive phi2
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    apply andb_prop in Hwfpphi.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: well_formed_positive phi1 = true
∧ well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi' destruct Hwfpphi as [Hwfpphi1 Hwfpphi2].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi1: ∀ n : nat,
  well_formed_positive phi1
  → (no_negative_occurrence_db_b n phi1
     → well_formed_positive phi1^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi1
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi1^[svar:n↦psi] &&
             well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    specialize (IHphi1 n' Hwfpphi1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: ∀ n : nat,
  well_formed_positive phi2
  → (no_negative_occurrence_db_b n phi2
     → well_formed_positive phi2^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi2
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi2^[svar:n↦psi] &&
             well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    specialize (IHphi2 n' Hwfpphi2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi1: (no_negative_occurrence_db_b n' phi1
 → well_formed_positive phi1^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi1
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi1^[svar:n'↦psi] &&
         well_formed_positive phi')
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    destruct IHphi1 as [IHphi11 IHphi12].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi2: (no_negative_occurrence_db_b n' phi2
 → well_formed_positive phi2^[svar:n'↦psi])
∧ (no_positive_occurrence_db_b n' phi2
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi2^[svar:n'↦psi] &&
         well_formed_positive phi')
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    destruct IHphi2 as [IHphi21 IHphi22].
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: no_negative_occurrence_db_b n' phi1
→ well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    specialize (IHphi11 Hnopos1).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi12: no_positive_occurrence_db_b n' phi1
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi1^[svar:n'↦psi] &&
      well_formed_positive phi'
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi' clear IHphi12.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: no_positive_occurrence_db_b n' phi2
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi2^[svar:n'↦psi] &&
      well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    specialize (IHphi22 Hnoneg2).
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi21: no_negative_occurrence_db_b n' phi2
→ well_formed_positive phi2^[svar:n'↦psi]
IHphi22: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi' clear IHphi21.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi22: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi1^[svar:n'↦psi] &&
well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi'
    rewrite IHphi11.
Σ: Signature
phi1, phi2, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
n': nat
IHphi11: well_formed_positive phi1^[svar:n'↦psi]
IHphi22: ∀ phi' : Pattern,
  well_formed_positive phi'
  → well_formed_positive phi2^[svar:n'↦psi] &&
    well_formed_positive phi'
Hwfpphi1: well_formed_positive phi1 = true
Hwfpphi2: well_formed_positive phi2 = true
Hnopos1: (fix no_negative_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_app ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_negative_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_negative_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_negative_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 with no_positive_occurrence_db_b
   (dbi : db_index) (ϕ : Pattern) {struct ϕ} :
     bool :=
   match ϕ with
   | patt_bound_svar n =>
       if decide (n = dbi)
       then false
       else true
   | patt_app ϕ₁ ϕ₂ =>
       no_positive_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_imp ϕ₁ ϕ₂ =>
       no_negative_occurrence_db_b dbi ϕ₁ &&
       no_positive_occurrence_db_b dbi ϕ₂
   | patt_exists ϕ' =>
       no_positive_occurrence_db_b dbi ϕ'
   | patt_mu ϕ' =>
       no_positive_occurrence_db_b 
         (S dbi) ϕ'
   | _ => true
   end
 for
 no_negative_occurrence_db_b) n' phi1 = true
Hnoneg2: no_positive_occurrence_db_b n' phi2 = true
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi2^[svar:n'↦psi] &&
well_formed_positive phi' rewrite IHphi22; auto.
-
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi') apply andb_prop in Hwfpphi.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi: no_negative_occurrence_db_b 0 phi = true
∧ well_formed_positive phi = true
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi') destruct Hwfpphi as [Hwfpphi1 Hwfpphi2].
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
  pose proof (IHphi' := IHphi (S n') Hwfpphi2).
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi': (no_negative_occurrence_db_b (S n') phi
 → well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
  destruct IHphi' as [IHphi1' IHphi2'].
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
  assert (H: no_negative_occurrence_db_b 0 (bsvar_subst phi psi (S n'))).
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
  {
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] clear IHphi1' IHphi2'.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
    apply no_neg_occ_db_bsvar_subst; auto.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
0 < S n' lia.
  }
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
(no_negative_occurrence_db_b (S n') phi
 → no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
   well_formed_positive phi^[svar:S n'↦psi])
∧ (no_positive_occurrence_db_b (S n') phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → no_negative_occurrence_db_b 0
           phi^[svar:S n'↦psi] &&
         well_formed_positive phi^[svar:S n'↦psi] &&
         well_formed_positive phi')
  split.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
no_negative_occurrence_db_b (S n') phi
→ no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
  well_formed_positive phi^[svar:S n'↦psi]
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → no_negative_occurrence_db_b 0
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
  +
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
no_negative_occurrence_db_b (S n') phi
→ no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
  well_formed_positive phi^[svar:S n'↦psi] intros Hnonegphi.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnonegphi: no_negative_occurrence_db_b (S n') phi
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
well_formed_positive phi^[svar:S n'↦psi]
    specialize (IHphi1' Hnonegphi).
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnonegphi: no_negative_occurrence_db_b (S n') phi
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
well_formed_positive phi^[svar:S n'↦psi]
    rewrite IHphi1'.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnonegphi: no_negative_occurrence_db_b (S n') phi
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
true
    rewrite H.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnonegphi: no_negative_occurrence_db_b (S n') phi
true && true
    auto.
  +
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → no_negative_occurrence_db_b 0
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi^[svar:S n'↦psi] &&
      well_formed_positive phi' intros Hnopos phi' Hwfpphi'.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi] &&
well_formed_positive phi^[svar:S n'↦psi] &&
well_formed_positive phi'
    rewrite H.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true && well_formed_positive phi^[svar:S n'↦psi] &&
well_formed_positive phi'
    rewrite IHphi2'.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
no_positive_occurrence_db_b (S n') phi
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi'
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true
    rewrite Hnopos.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
well_formed_positive phi'
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true
    2: rewrite Hwfpphi'.
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true
Σ: Signature
phi, psi: Pattern
Hwfppsi: well_formed_positive psi
Hwfcpsi: well_formed_closed_mu_aux psi 0
IHphi: ∀ n : nat,
  well_formed_positive phi
  → (no_negative_occurrence_db_b n phi
     → well_formed_positive phi^[svar:n↦psi])
    ∧ (no_positive_occurrence_db_b n phi
       → ∀ phi' : Pattern,
           well_formed_positive phi'
           → well_formed_positive
               phi^[svar:n↦psi] &&
             well_formed_positive phi')
n': nat
Hwfpphi1: no_negative_occurrence_db_b 0 phi = true
Hwfpphi2: well_formed_positive phi = true
IHphi1': no_negative_occurrence_db_b (S n') phi
→ well_formed_positive phi^[svar:S n'↦psi]
IHphi2': no_positive_occurrence_db_b (S n') phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        phi^[svar:S n'↦psi] &&
      well_formed_positive phi'
H: no_negative_occurrence_db_b 0 phi^[svar:S n'↦psi]
Hnopos: no_positive_occurrence_db_b (S n') phi
phi': Pattern
Hwfpphi': well_formed_positive phi'
true
    1,2,3: auto.
Qed.

Corollary wfp_bsvar_subst (phi psi : Pattern) :
well_formed_positive (patt_mu phi) ->
well_formed_positive psi ->
well_formed_closed_mu_aux psi 0 ->
well_formed_positive (bsvar_subst phi psi 0).
Σ: Signature
phi, psi: Pattern
well_formed_positive (patt_mu phi)
→ well_formed_positive psi
  → well_formed_closed_mu_aux psi 0
    → well_formed_positive phi^[svar:0↦psi]
Proof.
Σ: Signature
phi, psi: Pattern
well_formed_positive (patt_mu phi)
→ well_formed_positive psi
  → well_formed_closed_mu_aux psi 0
    → well_formed_positive phi^[svar:0↦psi]
intros H1 H2 H3.
Σ: Signature
phi, psi: Pattern
H1: well_formed_positive (patt_mu phi)
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
well_formed_positive phi^[svar:0↦psi]
simpl in H1.
Σ: Signature
phi, psi: Pattern
H1: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
well_formed_positive phi^[svar:0↦psi]
eapply elimT in H1.
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
H1: ?P
well_formed_positive phi^[svar:0↦psi]
Σ: Signature
phi, psi: Pattern
H1: no_negative_occurrence_db_b 0 phi &&
well_formed_positive phi
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
ssrbool.reflect ?P
  (no_negative_occurrence_db_b 0 phi &&
   well_formed_positive phi) 2: apply andP.
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
H1: no_negative_occurrence_db_b 0 phi
∧ well_formed_positive phi
well_formed_positive phi^[svar:0↦psi]
destruct H1 as [Hnonegphi Hwfpphi].
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
Hnonegphi: no_negative_occurrence_db_b 0 phi
Hwfpphi: well_formed_positive phi
well_formed_positive phi^[svar:0↦psi]
pose proof (H4 := Private_wfp_bsvar_subst).
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
Hnonegphi: no_negative_occurrence_db_b 0 phi
Hwfpphi: well_formed_positive phi
H4: ∀ (phi psi : Pattern) (n : nat),
  well_formed_positive psi
  → well_formed_closed_mu_aux psi 0
    → well_formed_positive phi
      → (no_negative_occurrence_db_b n phi
         → well_formed_positive phi^[svar:n↦psi])
        ∧ (no_positive_occurrence_db_b n phi
           → ∀ phi' : Pattern,
               well_formed_positive phi'
               → well_formed_positive
                   (patt_imp phi^[svar:n↦psi]
                     phi'))
well_formed_positive phi^[svar:0↦psi]
specialize (H4 phi psi 0 H2 H3 Hwfpphi).
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
Hnonegphi: no_negative_occurrence_db_b 0 phi
Hwfpphi: well_formed_positive phi
H4: (no_negative_occurrence_db_b 0 phi
 → well_formed_positive phi^[svar:0↦psi])
∧ (no_positive_occurrence_db_b 0 phi
   → ∀ phi' : Pattern,
       well_formed_positive phi'
       → well_formed_positive
           (patt_imp phi^[svar:0↦psi] phi'))
well_formed_positive phi^[svar:0↦psi]
destruct H4 as [H41 H42].
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
Hnonegphi: no_negative_occurrence_db_b 0 phi
Hwfpphi: well_formed_positive phi
H41: no_negative_occurrence_db_b 0 phi
→ well_formed_positive phi^[svar:0↦psi]
H42: no_positive_occurrence_db_b 0 phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        (patt_imp phi^[svar:0↦psi] phi')
well_formed_positive phi^[svar:0↦psi]
apply H41.
Σ: Signature
phi, psi: Pattern
H2: well_formed_positive psi
H3: well_formed_closed_mu_aux psi 0
Hnonegphi: no_negative_occurrence_db_b 0 phi
Hwfpphi: well_formed_positive phi
H41: no_negative_occurrence_db_b 0 phi
→ well_formed_positive phi^[svar:0↦psi]
H42: no_positive_occurrence_db_b 0 phi
→ ∀ phi' : Pattern,
    well_formed_positive phi'
    → well_formed_positive
        (patt_imp phi^[svar:0↦psi] phi')
no_negative_occurrence_db_b 0 phi
apply Hnonegphi.
Qed.


End subst.

Lemma bevar_subst_evar_quantify_free_evar {Σ : Signature} x dbi ϕ:
  well_formed_closed_ex_aux ϕ dbi ->
  bevar_subst (evar_quantify x dbi ϕ) (patt_free_evar x) dbi  = ϕ.
Σ: Signature
x: evar
dbi: db_index
ϕ: Pattern
well_formed_closed_ex_aux ϕ dbi
→ (evar_quantify x dbi ϕ)^[evar:dbi↦patt_free_evar x] =
  ϕ
Proof.
Σ: Signature
x: evar
dbi: db_index
ϕ: Pattern
well_formed_closed_ex_aux ϕ dbi
→ (evar_quantify x dbi ϕ)^[evar:dbi↦patt_free_evar x] =
  ϕ
  move: dbi.
Σ: Signature
x: evar
ϕ: Pattern
∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ dbi
  → (evar_quantify x dbi ϕ)^[evar:dbi↦patt_free_evar x] =
    ϕ
  induction ϕ; intros dbi Hwf; simpl in *; auto.
Σ: Signature
x, x0: evar
dbi: db_index
Hwf: true
(if decide (x = x0)
 then patt_bound_evar dbi
 else patt_free_evar x0)^[evar:dbi↦patt_free_evar x] =
patt_free_evar x0
Σ: Signature
x: evar
n, dbi: db_index
Hwf: if decide (n < dbi) then true else false
match compare_nat n dbi with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → (evar_quantify x dbi ϕ1)^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → (evar_quantify x dbi ϕ2)^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi
patt_app
  (evar_quantify x dbi ϕ1)^[evar:dbi↦patt_free_evar x]
  (evar_quantify x dbi ϕ2)^[evar:dbi↦patt_free_evar x] =
patt_app ϕ1 ϕ2
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → (evar_quantify x dbi ϕ1)^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → (evar_quantify x dbi ϕ2)^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi
patt_imp
  (evar_quantify x dbi ϕ1)^[evar:dbi↦
  patt_free_evar x]
  (evar_quantify x dbi ϕ2)^[evar:dbi↦
  patt_free_evar x] = patt_imp ϕ1 ϕ2
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ dbi
  → (evar_quantify x dbi ϕ)^[evar:dbi↦
    patt_free_evar x] = ϕ
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ (S dbi)
patt_exists
  (evar_quantify x (S dbi) ϕ)^[evar:
  S dbi↦patt_free_evar x] = 
patt_exists ϕ
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ dbi
  → (evar_quantify x dbi ϕ)^[evar:dbi↦
    patt_free_evar x] = ϕ
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ dbi
patt_mu
  (evar_quantify x dbi ϕ)^[evar:dbi↦
  patt_free_evar x] = patt_mu ϕ
  -
Σ: Signature
x, x0: evar
dbi: db_index
Hwf: true
(if decide (x = x0)
 then patt_bound_evar dbi
 else patt_free_evar x0)^[evar:dbi↦
patt_free_evar x] = patt_free_evar x0 case_match; simpl;[|reflexivity].
Σ: Signature
x, x0: evar
dbi: db_index
Hwf: true
e: x = x0
H: decide (x = x0) = left e
match compare_nat dbi dbi with
| Nat_less _ _ _ => patt_bound_evar dbi
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred dbi)
end = patt_free_evar x0
    case_match; simpl; try lia.
Σ: Signature
x, x0: evar
dbi: db_index
Hwf: true
e: x = x0
H: decide (x = x0) = left e
e0: dbi = dbi
H0: compare_nat dbi dbi = Nat_equal dbi dbi e0
patt_free_evar x = patt_free_evar x0 subst.
Σ: Signature
x0: evar
dbi: db_index
Hwf: true
H: decide (x0 = x0) = left (erefl x0)
e0: dbi = dbi
H0: compare_nat dbi dbi = Nat_equal dbi dbi e0
patt_free_evar x0 = patt_free_evar x0 auto.
  -
Σ: Signature
x: evar
n, dbi: db_index
Hwf: if decide (n < dbi) then true else false
match compare_nat n dbi with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end = patt_bound_evar n do 2 case_match; try lia; auto.
Σ: Signature
x: evar
n, dbi: db_index
n0: ¬ n < dbi
H: decide (n < dbi) = right n0
Hwf: false
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
patt_free_evar x = patt_bound_evar n
Σ: Signature
x: evar
n, dbi: db_index
n0: ¬ n < dbi
H: decide (n < dbi) = right n0
Hwf: false
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
patt_bound_evar (Nat.pred n) = patt_bound_evar n congruence.
Σ: Signature
x: evar
n, dbi: db_index
n0: ¬ n < dbi
H: decide (n < dbi) = right n0
Hwf: false
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
patt_bound_evar (Nat.pred n) = patt_bound_evar n congruence.
  -
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → (evar_quantify x dbi ϕ1)^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → (evar_quantify x dbi ϕ2)^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi
patt_app
  (evar_quantify x dbi ϕ1)^[evar:dbi↦
  patt_free_evar x]
  (evar_quantify x dbi ϕ2)^[evar:dbi↦
  patt_free_evar x] = patt_app ϕ1 ϕ2 apply andb_true_iff in Hwf.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → (evar_quantify x dbi ϕ1)^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → (evar_quantify x dbi ϕ2)^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ1 dbi = true
∧ well_formed_closed_ex_aux ϕ2 dbi = true
patt_app
  (evar_quantify x dbi ϕ1)^[evar:dbi↦
  patt_free_evar x]
  (evar_quantify x dbi ϕ2)^[evar:dbi↦
  patt_free_evar x] = patt_app ϕ1 ϕ2
    destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → (evar_quantify x dbi ϕ1)^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → (evar_quantify x dbi ϕ2)^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
H: well_formed_closed_ex_aux ϕ1 dbi = true
H0: well_formed_closed_ex_aux ϕ2 dbi = true
patt_app
  (evar_quantify x dbi ϕ1)^[evar:dbi↦
  patt_free_evar x]
  (evar_quantify x dbi ϕ2)^[evar:dbi↦
  patt_free_evar x] = patt_app ϕ1 ϕ2
    rewrite IHϕ1;[assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → (evar_quantify x dbi ϕ1)^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → (evar_quantify x dbi ϕ2)^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
H: well_formed_closed_ex_aux ϕ1 dbi = true
H0: well_formed_closed_ex_aux ϕ2 dbi = true
patt_app ϕ1
  (evar_quantify x dbi ϕ2)^[evar:dbi↦
  patt_free_evar x] = patt_app ϕ1 ϕ2
    rewrite IHϕ2;[assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → (evar_quantify x dbi ϕ1)^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → (evar_quantify x dbi ϕ2)^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
H: well_formed_closed_ex_aux ϕ1 dbi = true
H0: well_formed_closed_ex_aux ϕ2 dbi = true
patt_app ϕ1 ϕ2 = patt_app ϕ1 ϕ2
    reflexivity.
  -
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → (evar_quantify x dbi ϕ1)^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → (evar_quantify x dbi ϕ2)^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ1 dbi &&
well_formed_closed_ex_aux ϕ2 dbi
patt_imp
  (evar_quantify x dbi ϕ1)^[evar:dbi↦
  patt_free_evar x]
  (evar_quantify x dbi ϕ2)^[evar:dbi↦
  patt_free_evar x] = patt_imp ϕ1 ϕ2 apply andb_true_iff in Hwf.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → (evar_quantify x dbi ϕ1)^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → (evar_quantify x dbi ϕ2)^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ1 dbi = true
∧ well_formed_closed_ex_aux ϕ2 dbi = true
patt_imp
  (evar_quantify x dbi ϕ1)^[evar:dbi↦
  patt_free_evar x]
  (evar_quantify x dbi ϕ2)^[evar:dbi↦
  patt_free_evar x] = patt_imp ϕ1 ϕ2
    destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → (evar_quantify x dbi ϕ1)^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → (evar_quantify x dbi ϕ2)^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
H: well_formed_closed_ex_aux ϕ1 dbi = true
H0: well_formed_closed_ex_aux ϕ2 dbi = true
patt_imp
  (evar_quantify x dbi ϕ1)^[evar:dbi↦
  patt_free_evar x]
  (evar_quantify x dbi ϕ2)^[evar:dbi↦
  patt_free_evar x] = patt_imp ϕ1 ϕ2
    rewrite IHϕ1;[assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → (evar_quantify x dbi ϕ1)^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → (evar_quantify x dbi ϕ2)^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
H: well_formed_closed_ex_aux ϕ1 dbi = true
H0: well_formed_closed_ex_aux ϕ2 dbi = true
patt_imp ϕ1
  (evar_quantify x dbi ϕ2)^[evar:dbi↦
  patt_free_evar x] = patt_imp ϕ1 ϕ2
    rewrite IHϕ2;[assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ1 dbi
  → (evar_quantify x dbi ϕ1)^[evar:dbi↦
    patt_free_evar x] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ2 dbi
  → (evar_quantify x dbi ϕ2)^[evar:dbi↦
    patt_free_evar x] = ϕ2
dbi: db_index
H: well_formed_closed_ex_aux ϕ1 dbi = true
H0: well_formed_closed_ex_aux ϕ2 dbi = true
patt_imp ϕ1 ϕ2 = patt_imp ϕ1 ϕ2
    reflexivity.
  -
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ dbi
  → (evar_quantify x dbi ϕ)^[evar:dbi↦
    patt_free_evar x] = ϕ
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ (S dbi)
patt_exists
  (evar_quantify x (S dbi) ϕ)^[evar:
  S dbi↦patt_free_evar x] = 
patt_exists ϕ rewrite IHϕ;[assumption|reflexivity].
  -
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
  well_formed_closed_ex_aux ϕ dbi
  → (evar_quantify x dbi ϕ)^[evar:dbi↦
    patt_free_evar x] = ϕ
dbi: db_index
Hwf: well_formed_closed_ex_aux ϕ dbi
patt_mu
  (evar_quantify x dbi ϕ)^[evar:dbi↦
  patt_free_evar x] = patt_mu ϕ rewrite IHϕ;[assumption|reflexivity].
Qed.

Lemma bsvar_subst_svar_quantify_free_svar {Σ : Signature} X dbi ϕ:
  well_formed_closed_mu_aux ϕ dbi ->
  bsvar_subst (svar_quantify X dbi ϕ) (patt_free_svar X) dbi  = ϕ.
Σ: Signature
X: svar
dbi: db_index
ϕ: Pattern
well_formed_closed_mu_aux ϕ dbi
→ (svar_quantify X dbi ϕ)^[svar:dbi↦patt_free_svar X] =
  ϕ
Proof.
Σ: Signature
X: svar
dbi: db_index
ϕ: Pattern
well_formed_closed_mu_aux ϕ dbi
→ (svar_quantify X dbi ϕ)^[svar:dbi↦patt_free_svar X] =
  ϕ
  move: dbi.
Σ: Signature
X: svar
ϕ: Pattern
∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ dbi
  → (svar_quantify X dbi ϕ)^[svar:dbi↦patt_free_svar X] =
    ϕ
  induction ϕ; intros dbi Hwf; simpl in *; auto.
Σ: Signature
X, x: svar
dbi: db_index
Hwf: true
(if decide (X = x)
 then patt_bound_svar dbi
 else patt_free_svar x)^[svar:dbi↦patt_free_svar X] =
patt_free_svar x
Σ: Signature
X: svar
n, dbi: db_index
Hwf: if decide (n < dbi) then true else false
match compare_nat n dbi with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar X
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → (svar_quantify X dbi ϕ1)^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → (svar_quantify X dbi ϕ2)^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi
patt_app
  (svar_quantify X dbi ϕ1)^[svar:dbi↦patt_free_svar X]
  (svar_quantify X dbi ϕ2)^[svar:dbi↦patt_free_svar X] =
patt_app ϕ1 ϕ2
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → (svar_quantify X dbi ϕ1)^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → (svar_quantify X dbi ϕ2)^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi
patt_imp
  (svar_quantify X dbi ϕ1)^[svar:dbi↦
  patt_free_svar X]
  (svar_quantify X dbi ϕ2)^[svar:dbi↦
  patt_free_svar X] = patt_imp ϕ1 ϕ2
Σ: Signature
X: svar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ dbi
  → (svar_quantify X dbi ϕ)^[svar:dbi↦
    patt_free_svar X] = ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ dbi
patt_exists
  (svar_quantify X dbi ϕ)^[svar:dbi↦
  patt_free_svar X] = patt_exists ϕ
Σ: Signature
X: svar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ dbi
  → (svar_quantify X dbi ϕ)^[svar:dbi↦
    patt_free_svar X] = ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ (S dbi)
patt_mu
  (svar_quantify X (S dbi) ϕ)^[svar:
  S dbi↦patt_free_svar X] = 
patt_mu ϕ
  -
Σ: Signature
X, x: svar
dbi: db_index
Hwf: true
(if decide (X = x)
 then patt_bound_svar dbi
 else patt_free_svar x)^[svar:dbi↦
patt_free_svar X] = patt_free_svar x case_match; simpl;[|reflexivity].
Σ: Signature
X, x: svar
dbi: db_index
Hwf: true
e: X = x
H: decide (X = x) = left e
match compare_nat dbi dbi with
| Nat_less _ _ _ => patt_bound_svar dbi
| Nat_equal _ _ _ => patt_free_svar X
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred dbi)
end = patt_free_svar x
    case_match; simpl; try lia.
Σ: Signature
X, x: svar
dbi: db_index
Hwf: true
e: X = x
H: decide (X = x) = left e
e0: dbi = dbi
H0: compare_nat dbi dbi = Nat_equal dbi dbi e0
patt_free_svar X = patt_free_svar x subst.
Σ: Signature
x: svar
dbi: db_index
Hwf: true
H: decide (x = x) = left (erefl x)
e0: dbi = dbi
H0: compare_nat dbi dbi = Nat_equal dbi dbi e0
patt_free_svar x = patt_free_svar x auto.
  -
Σ: Signature
X: svar
n, dbi: db_index
Hwf: if decide (n < dbi) then true else false
match compare_nat n dbi with
| Nat_less _ _ _ => patt_bound_svar n
| Nat_equal _ _ _ => patt_free_svar X
| Nat_greater _ _ _ => patt_bound_svar (Nat.pred n)
end = patt_bound_svar n do 2 case_match; try lia; auto.
Σ: Signature
X: svar
n, dbi: db_index
n0: ¬ n < dbi
H: decide (n < dbi) = right n0
Hwf: false
e: n = dbi
H0: compare_nat n dbi = Nat_equal n dbi e
patt_free_svar X = patt_bound_svar n
Σ: Signature
X: svar
n, dbi: db_index
n0: ¬ n < dbi
H: decide (n < dbi) = right n0
Hwf: false
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
patt_bound_svar (Nat.pred n) = patt_bound_svar n congruence.
Σ: Signature
X: svar
n, dbi: db_index
n0: ¬ n < dbi
H: decide (n < dbi) = right n0
Hwf: false
g: n > dbi
H0: compare_nat n dbi = Nat_greater n dbi g
patt_bound_svar (Nat.pred n) = patt_bound_svar n congruence.
  -
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → (svar_quantify X dbi ϕ1)^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → (svar_quantify X dbi ϕ2)^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi
patt_app
  (svar_quantify X dbi ϕ1)^[svar:dbi↦
  patt_free_svar X]
  (svar_quantify X dbi ϕ2)^[svar:dbi↦
  patt_free_svar X] = patt_app ϕ1 ϕ2 apply andb_true_iff in Hwf.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → (svar_quantify X dbi ϕ1)^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → (svar_quantify X dbi ϕ2)^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ1 dbi = true
∧ well_formed_closed_mu_aux ϕ2 dbi = true
patt_app
  (svar_quantify X dbi ϕ1)^[svar:dbi↦
  patt_free_svar X]
  (svar_quantify X dbi ϕ2)^[svar:dbi↦
  patt_free_svar X] = patt_app ϕ1 ϕ2
    destruct_and!.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → (svar_quantify X dbi ϕ1)^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → (svar_quantify X dbi ϕ2)^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
H: well_formed_closed_mu_aux ϕ1 dbi = true
H0: well_formed_closed_mu_aux ϕ2 dbi = true
patt_app
  (svar_quantify X dbi ϕ1)^[svar:dbi↦
  patt_free_svar X]
  (svar_quantify X dbi ϕ2)^[svar:dbi↦
  patt_free_svar X] = patt_app ϕ1 ϕ2
    rewrite IHϕ1;[assumption|].
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → (svar_quantify X dbi ϕ1)^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → (svar_quantify X dbi ϕ2)^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
H: well_formed_closed_mu_aux ϕ1 dbi = true
H0: well_formed_closed_mu_aux ϕ2 dbi = true
patt_app ϕ1
  (svar_quantify X dbi ϕ2)^[svar:dbi↦
  patt_free_svar X] = patt_app ϕ1 ϕ2
    rewrite IHϕ2;[assumption|].
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → (svar_quantify X dbi ϕ1)^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → (svar_quantify X dbi ϕ2)^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
H: well_formed_closed_mu_aux ϕ1 dbi = true
H0: well_formed_closed_mu_aux ϕ2 dbi = true
patt_app ϕ1 ϕ2 = patt_app ϕ1 ϕ2
    reflexivity.
  -
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → (svar_quantify X dbi ϕ1)^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → (svar_quantify X dbi ϕ2)^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ1 dbi &&
well_formed_closed_mu_aux ϕ2 dbi
patt_imp
  (svar_quantify X dbi ϕ1)^[svar:dbi↦
  patt_free_svar X]
  (svar_quantify X dbi ϕ2)^[svar:dbi↦
  patt_free_svar X] = patt_imp ϕ1 ϕ2 apply andb_true_iff in Hwf.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → (svar_quantify X dbi ϕ1)^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → (svar_quantify X dbi ϕ2)^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ1 dbi = true
∧ well_formed_closed_mu_aux ϕ2 dbi = true
patt_imp
  (svar_quantify X dbi ϕ1)^[svar:dbi↦
  patt_free_svar X]
  (svar_quantify X dbi ϕ2)^[svar:dbi↦
  patt_free_svar X] = patt_imp ϕ1 ϕ2
    destruct_and!.
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → (svar_quantify X dbi ϕ1)^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → (svar_quantify X dbi ϕ2)^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
H: well_formed_closed_mu_aux ϕ1 dbi = true
H0: well_formed_closed_mu_aux ϕ2 dbi = true
patt_imp
  (svar_quantify X dbi ϕ1)^[svar:dbi↦
  patt_free_svar X]
  (svar_quantify X dbi ϕ2)^[svar:dbi↦
  patt_free_svar X] = patt_imp ϕ1 ϕ2
    rewrite IHϕ1;[assumption|].
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → (svar_quantify X dbi ϕ1)^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → (svar_quantify X dbi ϕ2)^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
H: well_formed_closed_mu_aux ϕ1 dbi = true
H0: well_formed_closed_mu_aux ϕ2 dbi = true
patt_imp ϕ1
  (svar_quantify X dbi ϕ2)^[svar:dbi↦
  patt_free_svar X] = patt_imp ϕ1 ϕ2
    rewrite IHϕ2;[assumption|].
Σ: Signature
X: svar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ1 dbi
  → (svar_quantify X dbi ϕ1)^[svar:dbi↦
    patt_free_svar X] = ϕ1
IHϕ2: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ2 dbi
  → (svar_quantify X dbi ϕ2)^[svar:dbi↦
    patt_free_svar X] = ϕ2
dbi: db_index
H: well_formed_closed_mu_aux ϕ1 dbi = true
H0: well_formed_closed_mu_aux ϕ2 dbi = true
patt_imp ϕ1 ϕ2 = patt_imp ϕ1 ϕ2
    reflexivity.
  -
Σ: Signature
X: svar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ dbi
  → (svar_quantify X dbi ϕ)^[svar:dbi↦
    patt_free_svar X] = ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ dbi
patt_exists
  (svar_quantify X dbi ϕ)^[svar:dbi↦
  patt_free_svar X] = patt_exists ϕ rewrite IHϕ;[assumption|reflexivity].
  -
Σ: Signature
X: svar
ϕ: Pattern
IHϕ: ∀ dbi : db_index,
  well_formed_closed_mu_aux ϕ dbi
  → (svar_quantify X dbi ϕ)^[svar:dbi↦
    patt_free_svar X] = ϕ
dbi: db_index
Hwf: well_formed_closed_mu_aux ϕ (S dbi)
patt_mu
  (svar_quantify X (S dbi) ϕ)^[svar:
  S dbi↦patt_free_svar X] = 
patt_mu ϕ rewrite IHϕ;[assumption|reflexivity].
Qed.



Lemma free_svar_subst_fresh {Σ : Signature} phi psi X:
  svar_is_fresh_in X phi ->
  free_svar_subst phi psi X = phi.
Σ: Signature
phi, psi: Pattern
X: svar
svar_is_fresh_in X phi → phi^[[svar:X↦psi]] = phi
Proof.
Σ: Signature
phi, psi: Pattern
X: svar
svar_is_fresh_in X phi → phi^[[svar:X↦psi]] = phi
  intros Hfresh.
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: svar_is_fresh_in X phi
phi^[[svar:X↦psi]] = phi
  unfold svar_is_fresh_in in Hfresh.
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
phi^[[svar:X↦psi]] = phi
  induction phi; simpl in *; auto.
Σ: Signature
x: svar
psi: Pattern
X: svar
Hfresh: X ∉ {[x]}
(if decide (X = x) then psi else patt_free_svar x) =
patt_free_svar x
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: X ∉ free_svars phi1
→ phi1^[[svar:X↦psi]] = phi1
IHphi2: X ∉ free_svars phi2
→ phi2^[[svar:X↦psi]] = phi2
patt_app phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_app phi1 phi2
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: X ∉ free_svars phi1
→ phi1^[[svar:X↦psi]] = phi1
IHphi2: X ∉ free_svars phi2
→ phi2^[[svar:X↦psi]] = phi2
patt_imp phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_imp phi1 phi2
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: X ∉ free_svars phi → phi^[[svar:X↦psi]] = phi
patt_exists phi^[[svar:X↦psi]] = patt_exists phi
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: X ∉ free_svars phi → phi^[[svar:X↦psi]] = phi
patt_mu phi^[[svar:X↦psi]] = patt_mu phi
  -
Σ: Signature
x: svar
psi: Pattern
X: svar
Hfresh: X ∉ {[x]}
(if decide (X = x) then psi else patt_free_svar x) =
patt_free_svar x case_match.
Σ: Signature
x: svar
psi: Pattern
X: svar
Hfresh: X ∉ {[x]}
e: X = x
H: decide (X = x) = left e
psi = patt_free_svar x
Σ: Signature
x: svar
psi: Pattern
X: svar
Hfresh: X ∉ {[x]}
n: X ≠ x
H: decide (X = x) = right n
patt_free_svar x = patt_free_svar x
    +
Σ: Signature
x: svar
psi: Pattern
X: svar
Hfresh: X ∉ {[x]}
e: X = x
H: decide (X = x) = left e
psi = patt_free_svar x subst.
Σ: Signature
x: svar
psi: Pattern
Hfresh: x ∉ {[x]}
H: decide (x = x) = left (erefl x)
psi = patt_free_svar x set_solver.
    +
Σ: Signature
x: svar
psi: Pattern
X: svar
Hfresh: X ∉ {[x]}
n: X ≠ x
H: decide (X = x) = right n
patt_free_svar x = patt_free_svar x reflexivity.
  -
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: X ∉ free_svars phi1
→ phi1^[[svar:X↦psi]] = phi1
IHphi2: X ∉ free_svars phi2
→ phi2^[[svar:X↦psi]] = phi2
patt_app phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_app phi1 phi2 specialize (IHphi1 ltac:(set_solver)).
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: X ∉ free_svars phi2
→ phi2^[[svar:X↦psi]] = phi2
patt_app phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_app phi1 phi2
    specialize (IHphi2 ltac:(set_solver)).
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: phi2^[[svar:X↦psi]] = phi2
patt_app phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_app phi1 phi2
    rewrite IHphi1.
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: phi2^[[svar:X↦psi]] = phi2
patt_app phi1 phi2^[[svar:X↦psi]] = patt_app phi1 phi2 rewrite IHphi2.
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: phi2^[[svar:X↦psi]] = phi2
patt_app phi1 phi2 = patt_app phi1 phi2
    reflexivity.
  -
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: X ∉ free_svars phi1
→ phi1^[[svar:X↦psi]] = phi1
IHphi2: X ∉ free_svars phi2
→ phi2^[[svar:X↦psi]] = phi2
patt_imp phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_imp phi1 phi2 specialize (IHphi1 ltac:(set_solver)).
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: X ∉ free_svars phi2
→ phi2^[[svar:X↦psi]] = phi2
patt_imp phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_imp phi1 phi2
    specialize (IHphi2 ltac:(set_solver)).
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: phi2^[[svar:X↦psi]] = phi2
patt_imp phi1^[[svar:X↦psi]] phi2^[[svar:X↦psi]] =
patt_imp phi1 phi2
    rewrite IHphi1.
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: phi2^[[svar:X↦psi]] = phi2
patt_imp phi1 phi2^[[svar:X↦psi]] = patt_imp phi1 phi2 rewrite IHphi2.
Σ: Signature
phi1, phi2, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi1 ∪ free_svars phi2
IHphi1: phi1^[[svar:X↦psi]] = phi1
IHphi2: phi2^[[svar:X↦psi]] = phi2
patt_imp phi1 phi2 = patt_imp phi1 phi2
    reflexivity.
  -
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: X ∉ free_svars phi → phi^[[svar:X↦psi]] = phi
patt_exists phi^[[svar:X↦psi]] = patt_exists phi specialize (IHphi ltac:(assumption)).
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: phi^[[svar:X↦psi]] = phi
patt_exists phi^[[svar:X↦psi]] = patt_exists phi
    rewrite IHphi.
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: phi^[[svar:X↦psi]] = phi
patt_exists phi = patt_exists phi
    reflexivity.
  -
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: X ∉ free_svars phi → phi^[[svar:X↦psi]] = phi
patt_mu phi^[[svar:X↦psi]] = patt_mu phi specialize (IHphi ltac:(assumption)).
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: phi^[[svar:X↦psi]] = phi
patt_mu phi^[[svar:X↦psi]] = patt_mu phi
    rewrite IHphi.
Σ: Signature
phi, psi: Pattern
X: svar
Hfresh: X ∉ free_svars phi
IHphi: phi^[[svar:X↦psi]] = phi
patt_mu phi = patt_mu phi
    reflexivity.
Qed.


Lemma wfc_mu_free_svar_subst {Σ : Signature} level ϕ ψ X:
  well_formed_closed_mu_aux ϕ level ->
  well_formed_closed_mu_aux ψ level ->
  well_formed_closed_mu_aux (free_svar_subst ϕ ψ X) level = true.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
  → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] level =
    true
Proof.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
  → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] level =
    true
  intros Hϕ Hψ.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: svar
Hϕ: well_formed_closed_mu_aux ϕ level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] level = true
  move: level Hϕ Hψ.
Σ: Signature
ϕ, ψ: Pattern
X: svar
∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] level =
      true
  induction ϕ; intros level Hϕ Hψ; simpl in *; auto.
Σ: Signature
x: svar
ψ: Pattern
X: svar
level: db_index
Hϕ: true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux
  (if decide (X = x) then ψ else patt_free_svar x)
  level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] (S level) =
true
  -
Σ: Signature
x: svar
ψ: Pattern
X: svar
level: db_index
Hϕ: true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux
  (if decide (X = x) then ψ else patt_free_svar x)
  level = true case_match; [|reflexivity].
Σ: Signature
x: svar
ψ: Pattern
X: svar
level: db_index
Hϕ: true
Hψ: well_formed_closed_mu_aux ψ level
e: X = x
H: decide (X = x) = left e
well_formed_closed_mu_aux ψ level = true
    assumption.
  -
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ1; auto.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
true &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ2; auto.
  -
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ1; auto.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
true &&
well_formed_closed_mu_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ2; auto.
  -
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]] (S level) =
true rewrite IHϕ; auto.
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ψ (S level) eapply well_formed_closed_mu_aux_ind.
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
?ind_svar1 ≤ S level
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ψ ?ind_svar1 = true 2: exact Hψ.
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
level ≤ S level lia.
Qed.

#[export]
 Hint Resolve wfc_mu_free_svar_subst : core.

Lemma wfc_ex_free_svar_subst {Σ : Signature} level ϕ ψ X:
  well_formed_closed_ex_aux ϕ level ->
  well_formed_closed_ex_aux ψ level ->
  well_formed_closed_ex_aux (free_svar_subst ϕ ψ X) level = true.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
  → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] level =
    true
Proof.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: svar
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
  → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] level =
    true
  intros Hϕ Hψ.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: svar
Hϕ: well_formed_closed_ex_aux ϕ level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] level = true
  move: level Hϕ Hψ.
Σ: Signature
ϕ, ψ: Pattern
X: svar
∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] level =
      true
  induction ϕ; intros level Hϕ Hψ; simpl in *; auto.
Σ: Signature
x: svar
ψ: Pattern
X: svar
level: db_index
Hϕ: true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux
  (if decide (X = x) then ψ else patt_free_svar x)
  level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] (S level) =
true
  -
Σ: Signature
x: svar
ψ: Pattern
X: svar
level: db_index
Hϕ: true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux
  (if decide (X = x) then ψ else patt_free_svar x)
  level = true case_match; [|reflexivity].
Σ: Signature
x: svar
ψ: Pattern
X: svar
level: db_index
Hϕ: true
Hψ: well_formed_closed_ex_aux ψ level
e: X = x
H: decide (X = x) = left e
well_formed_closed_ex_aux ψ level = true
    assumption.
  -
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ1; auto.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
true &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ2; auto.
  -
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ1; auto.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: svar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[svar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
true &&
well_formed_closed_ex_aux ϕ2^[[svar:X↦ψ]] level = true
    rewrite IHϕ2; auto.
  -
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]] (S level) =
true rewrite IHϕ; auto.
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ψ (S level) eapply well_formed_closed_ex_aux_ind.
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
?ind_evar1 ≤ S level
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ψ ?ind_evar1 = true 2: exact Hψ.
Σ: Signature
ϕ, ψ: Pattern
X: svar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[svar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
level ≤ S level lia.
Qed.

#[export]
 Hint Resolve wfc_mu_free_svar_subst : core.

Lemma wfc_ex_free_evar_subst_2 {Σ : Signature} level ϕ ψ X:
  well_formed_closed_ex_aux ϕ level ->
  well_formed_closed_ex_aux ψ level ->
  well_formed_closed_ex_aux (free_evar_subst ϕ ψ X) level = true.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: evar
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
  → well_formed_closed_ex_aux ϕ^[[evar:X↦ψ]] level =
    true
Proof.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: evar
well_formed_closed_ex_aux ϕ level
→ well_formed_closed_ex_aux ψ level
  → well_formed_closed_ex_aux ϕ^[[evar:X↦ψ]] level =
    true
  intros Hϕ Hψ.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
X: evar
Hϕ: well_formed_closed_ex_aux ϕ level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ^[[evar:X↦ψ]] level = true
  move: level Hϕ Hψ.
Σ: Signature
ϕ, ψ: Pattern
X: evar
∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[evar:X↦ψ]] level =
      true
  induction ϕ; intros level Hϕ Hψ; simpl in *; auto.
Σ: Signature
x: evar
ψ: Pattern
X: evar
level: db_index
Hϕ: true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux
  (if decide (X = x) then ψ else patt_free_evar x)
  level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]] level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]] level = true
Σ: Signature
ϕ, ψ: Pattern
X: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[evar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ^[[evar:X↦ψ]] (S level) =
true
  -
Σ: Signature
x: evar
ψ: Pattern
X: evar
level: db_index
Hϕ: true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux
  (if decide (X = x) then ψ else patt_free_evar x)
  level = true case_match; [|reflexivity].
Σ: Signature
x: evar
ψ: Pattern
X: evar
level: db_index
Hϕ: true
Hψ: well_formed_closed_ex_aux ψ level
e: X = x
H: decide (X = x) = left e
well_formed_closed_ex_aux ψ level = true
    repeat case_match; auto.
  -
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]] level = true
    rewrite IHϕ1; auto.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
true &&
well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]] level = true
    rewrite IHϕ2; auto.
  -
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ1 level &&
well_formed_closed_ex_aux ϕ2 level
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]] level &&
well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]] level = true
    rewrite IHϕ1; auto.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
X: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ1 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ1^[[evar:X↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ2 level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]]
        level = true
level: db_index
H: well_formed_closed_ex_aux ϕ1 level = true
H0: well_formed_closed_ex_aux ϕ2 level = true
Hψ: well_formed_closed_ex_aux ψ level
true &&
well_formed_closed_ex_aux ϕ2^[[evar:X↦ψ]] level = true
    rewrite IHϕ2; auto.
  -
Σ: Signature
ϕ, ψ: Pattern
X: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[evar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ϕ^[[evar:X↦ψ]] (S level) =
true rewrite IHϕ; auto.
Σ: Signature
ϕ, ψ: Pattern
X: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[evar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ψ (S level) eapply well_formed_closed_ex_aux_ind.
Σ: Signature
ϕ, ψ: Pattern
X: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[evar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
?ind_evar1 ≤ S level
Σ: Signature
ϕ, ψ: Pattern
X: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[evar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
well_formed_closed_ex_aux ψ ?ind_evar1 = true 2: exact Hψ.
Σ: Signature
ϕ, ψ: Pattern
X: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_ex_aux ϕ level
  → well_formed_closed_ex_aux ψ level
    → well_formed_closed_ex_aux ϕ^[[evar:X↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_ex_aux ϕ (S level)
Hψ: well_formed_closed_ex_aux ψ level
level ≤ S level lia.
Qed.

#[export]
 Hint Resolve wfc_ex_free_evar_subst_2 : core.


Lemma wfc_mu_free_evar_subst {Σ : Signature} level ϕ ψ x:
well_formed_closed_mu_aux ϕ level ->
well_formed_closed_mu_aux ψ level ->
well_formed_closed_mu_aux (free_evar_subst ϕ ψ x) level = true.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
x: evar
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
  → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] level =
    true
Proof.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
x: evar
well_formed_closed_mu_aux ϕ level
→ well_formed_closed_mu_aux ψ level
  → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] level =
    true
intros Hϕ Hψ.
Σ: Signature
level: db_index
ϕ, ψ: Pattern
x: evar
Hϕ: well_formed_closed_mu_aux ϕ level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] level = true
move: level Hϕ Hψ.
Σ: Signature
ϕ, ψ: Pattern
x: evar
∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] level =
      true
induction ϕ; intros level Hϕ Hψ; simpl in *; auto.
Σ: Signature
x0: evar
ψ: Pattern
x: evar
level: db_index
Hϕ: true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux
  (if decide (x = x0) then ψ else patt_free_evar x0)
  level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] (S level) =
true
-
Σ: Signature
x0: evar
ψ: Pattern
x: evar
level: db_index
Hϕ: true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux
  (if decide (x = x0) then ψ else patt_free_evar x0)
  level = true case_match; [|reflexivity].
Σ: Signature
x0: evar
ψ: Pattern
x: evar
level: db_index
Hϕ: true
Hψ: well_formed_closed_mu_aux ψ level
e: x = x0
H: decide (x = x0) = left e
well_formed_closed_mu_aux ψ level = true
  assumption.
-
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true
  rewrite IHϕ1; auto.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
true &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true
  rewrite IHϕ2; auto.
-
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ1 level &&
well_formed_closed_mu_aux ϕ2 level
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true destruct_and!.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]] level &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true
  rewrite IHϕ1; auto.
Σ: Signature
ϕ1, ϕ2, ψ: Pattern
x: evar
IHϕ1: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ1 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ1^[[evar:x↦ψ]]
        level = true
IHϕ2: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ2 level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]]
        level = true
level: db_index
H: well_formed_closed_mu_aux ϕ1 level = true
H0: well_formed_closed_mu_aux ϕ2 level = true
Hψ: well_formed_closed_mu_aux ψ level
true &&
well_formed_closed_mu_aux ϕ2^[[evar:x↦ψ]] level = true
  rewrite IHϕ2; auto.
-
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]] (S level) =
true apply IHϕ; auto.
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ψ (S level)
  eapply well_formed_closed_mu_aux_ind.
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
?ind_svar1 ≤ S level
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
well_formed_closed_mu_aux ψ ?ind_svar1 = true
  2: eassumption.
Σ: Signature
ϕ, ψ: Pattern
x: evar
IHϕ: ∀ level : db_index,
  well_formed_closed_mu_aux ϕ level
  → well_formed_closed_mu_aux ψ level
    → well_formed_closed_mu_aux ϕ^[[evar:x↦ψ]]
        level = true
level: db_index
Hϕ: well_formed_closed_mu_aux ϕ (S level)
Hψ: well_formed_closed_mu_aux ψ level
level ≤ S level lia.
Qed.


Lemma wf_evar_open_from_wf_ex {Σ : Signature} x ϕ:
  well_formed (patt_exists ϕ) ->
  well_formed (evar_open 0 x ϕ).
Σ: Signature
x: evar
ϕ: Pattern
well_formed (patt_exists ϕ)
→ well_formed (evar_open 0 x ϕ)
Proof.
Σ: Signature
x: evar
ϕ: Pattern
well_formed (patt_exists ϕ)
→ well_formed (evar_open 0 x ϕ)
  intros H.
Σ: Signature
x: evar
ϕ: Pattern
H: well_formed (patt_exists ϕ)
well_formed (evar_open 0 x ϕ)
  unfold well_formed, well_formed_closed in *.
Σ: Signature
x: evar
ϕ: Pattern
H: [&& well_formed_positive (patt_exists ϕ),
    well_formed_closed_mu_aux (patt_exists ϕ) 0
  & well_formed_closed_ex_aux (patt_exists ϕ) 0]
[&& well_formed_positive (evar_open 0 x ϕ),
    well_formed_closed_mu_aux (evar_open 0 x ϕ) 0
  & well_formed_closed_ex_aux (evar_open 0 x ϕ) 0]
  destruct_and!.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive (patt_exists ϕ) = true
H: well_formed_closed_mu_aux (patt_exists ϕ) 0 = true
H2: well_formed_closed_ex_aux (patt_exists ϕ) 0 =
true
[&& well_formed_positive (evar_open 0 x ϕ),
    well_formed_closed_mu_aux (evar_open 0 x ϕ) 0
  & well_formed_closed_ex_aux (evar_open 0 x ϕ) 0] cbn in *.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
[&& well_formed_positive (evar_open 0 x ϕ),
    well_formed_closed_mu_aux (evar_open 0 x ϕ) 0
  & well_formed_closed_ex_aux (evar_open 0 x ϕ) 0] split_and!.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_positive (evar_open 0 x ϕ) = true
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_mu_aux (evar_open 0 x ϕ) 0 = true
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_ex_aux (evar_open 0 x ϕ) 0 = true
  -
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_positive (evar_open 0 x ϕ) = true apply wfp_evar_open.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_positive ϕ = true assumption.
  -
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_mu_aux (evar_open 0 x ϕ) 0 = true apply wfc_mu_aux_body_ex_imp1.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_mu_aux ϕ 0 = true assumption.
  -
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_ex_aux (evar_open 0 x ϕ) 0 = true apply wfc_mu_aux_body_ex_imp3.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
0 ≤ 0
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_ex_aux ϕ 1 = true lia.
Σ: Signature
x: evar
ϕ: Pattern
H0: well_formed_positive ϕ = true
H: well_formed_closed_mu_aux ϕ 0 = true
H2: well_formed_closed_ex_aux ϕ 1 = true
well_formed_closed_ex_aux ϕ 1 = true assumption.
Qed.


Lemma evar_open_size' {Σ : Signature} :
  forall (k : db_index) (n : evar) (p : Pattern),
    size' (evar_open k n p) = size' p.
Σ: Signature
∀ (k : db_index) (n : evar) (p : Pattern),
  size' (evar_open k n p) = size' p
Proof.
Σ: Signature
∀ (k : db_index) (n : evar) (p : Pattern),
  size' (evar_open k n p) = size' p
  intros k n p.
Σ: Signature
k: db_index
n: evar
p: Pattern
size' (evar_open k n p) = size' p generalize dependent k.
Σ: Signature
n: evar
p: Pattern
∀ k : db_index, size' (evar_open k n p) = size' p
  induction p; intros k; cbn; try reflexivity.
Σ: Signature
n: evar
n0, k: db_index
size'
  match compare_nat n0 k with
  | Nat_less _ _ _ => patt_bound_evar n0
  | Nat_equal _ _ _ => patt_free_evar n
  | Nat_greater _ _ _ => patt_bound_evar (Nat.pred n0)
  end = 1
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size' (evar_open k n p1) = size' p1
IHp2: ∀ k : db_index,
  size' (evar_open k n p2) = size' p2
k: db_index
S
  (size' p1^[evar:k↦patt_free_evar n] +
   size' p2^[evar:k↦patt_free_evar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size' (evar_open k n p1) = size' p1
IHp2: ∀ k : db_index,
  size' (evar_open k n p2) = size' p2
k: db_index
S
  (size' p1^[evar:k↦patt_free_evar n] +
   size' p2^[evar:k↦patt_free_evar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index,
  size' (evar_open k n p) = size' p
k: db_index
S (size' p^[evar:S k↦patt_free_evar n]) = S (size' p)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index,
  size' (evar_open k n p) = size' p
k: db_index
S (size' p^[evar:k↦patt_free_evar n]) = S (size' p)
  break_match_goal; reflexivity.
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size' (evar_open k n p1) = size' p1
IHp2: ∀ k : db_index,
  size' (evar_open k n p2) = size' p2
k: db_index
S
  (size' p1^[evar:k↦patt_free_evar n] +
   size' p2^[evar:k↦patt_free_evar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size' (evar_open k n p1) = size' p1
IHp2: ∀ k : db_index,
  size' (evar_open k n p2) = size' p2
k: db_index
S
  (size' p1^[evar:k↦patt_free_evar n] +
   size' p2^[evar:k↦patt_free_evar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index,
  size' (evar_open k n p) = size' p
k: db_index
S (size' p^[evar:S k↦patt_free_evar n]) = S (size' p)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index,
  size' (evar_open k n p) = size' p
k: db_index
S (size' p^[evar:k↦patt_free_evar n]) = S (size' p)
  rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: evar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size' (evar_open k n p1) = size' p1
IHp2: ∀ k : db_index,
  size' (evar_open k n p2) = size' p2
k: db_index
S
  (size' p1^[evar:k↦patt_free_evar n] +
   size' p2^[evar:k↦patt_free_evar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index,
  size' (evar_open k n p) = size' p
k: db_index
S (size' p^[evar:S k↦patt_free_evar n]) = S (size' p)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index,
  size' (evar_open k n p) = size' p
k: db_index
S (size' p^[evar:k↦patt_free_evar n]) = S (size' p)
  rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index,
  size' (evar_open k n p) = size' p
k: db_index
S (size' p^[evar:S k↦patt_free_evar n]) = S (size' p)
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index,
  size' (evar_open k n p) = size' p
k: db_index
S (size' p^[evar:k↦patt_free_evar n]) = S (size' p)
  rewrite (IHp (S k)); reflexivity.
Σ: Signature
n: evar
p: Pattern
IHp: ∀ k : db_index,
  size' (evar_open k n p) = size' p
k: db_index
S (size' p^[evar:k↦patt_free_evar n]) = S (size' p)
  rewrite (IHp k); reflexivity.
Qed.

Lemma svar_open_size' {Σ : Signature} :
  forall (k : db_index) (n : svar) (p : Pattern),
    size' (svar_open k n p) = size' p.
Σ: Signature
∀ (k : db_index) (n : svar) (p : Pattern),
  size' (svar_open k n p) = size' p
Proof.
Σ: Signature
∀ (k : db_index) (n : svar) (p : Pattern),
  size' (svar_open k n p) = size' p
  intros k n p.
Σ: Signature
k: db_index
n: svar
p: Pattern
size' (svar_open k n p) = size' p generalize dependent k.
Σ: Signature
n: svar
p: Pattern
∀ k : db_index, size' (svar_open k n p) = size' p
  induction p; intros k; cbn; try reflexivity.
Σ: Signature
n: svar
n0, k: db_index
size'
  match compare_nat n0 k with
  | Nat_less _ _ _ => patt_bound_svar n0
  | Nat_equal _ _ _ => patt_free_svar n
  | Nat_greater _ _ _ => patt_bound_svar (Nat.pred n0)
  end = 1
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size' (svar_open k n p1) = size' p1
IHp2: ∀ k : db_index,
  size' (svar_open k n p2) = size' p2
k: db_index
S
  (size' p1^[svar:k↦patt_free_svar n] +
   size' p2^[svar:k↦patt_free_svar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size' (svar_open k n p1) = size' p1
IHp2: ∀ k : db_index,
  size' (svar_open k n p2) = size' p2
k: db_index
S
  (size' p1^[svar:k↦patt_free_svar n] +
   size' p2^[svar:k↦patt_free_svar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index,
  size' (svar_open k n p) = size' p
k: db_index
S (size' p^[svar:k↦patt_free_svar n]) = S (size' p)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index,
  size' (svar_open k n p) = size' p
k: db_index
S (size' p^[svar:S k↦patt_free_svar n]) = S (size' p)
  break_match_goal; reflexivity.
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size' (svar_open k n p1) = size' p1
IHp2: ∀ k : db_index,
  size' (svar_open k n p2) = size' p2
k: db_index
S
  (size' p1^[svar:k↦patt_free_svar n] +
   size' p2^[svar:k↦patt_free_svar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size' (svar_open k n p1) = size' p1
IHp2: ∀ k : db_index,
  size' (svar_open k n p2) = size' p2
k: db_index
S
  (size' p1^[svar:k↦patt_free_svar n] +
   size' p2^[svar:k↦patt_free_svar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index,
  size' (svar_open k n p) = size' p
k: db_index
S (size' p^[svar:k↦patt_free_svar n]) = S (size' p)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index,
  size' (svar_open k n p) = size' p
k: db_index
S (size' p^[svar:S k↦patt_free_svar n]) = S (size' p)
  rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: svar
p1, p2: Pattern
IHp1: ∀ k : db_index,
  size' (svar_open k n p1) = size' p1
IHp2: ∀ k : db_index,
  size' (svar_open k n p2) = size' p2
k: db_index
S
  (size' p1^[svar:k↦patt_free_svar n] +
   size' p2^[svar:k↦patt_free_svar n]) =
S (size' p1 + size' p2)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index,
  size' (svar_open k n p) = size' p
k: db_index
S (size' p^[svar:k↦patt_free_svar n]) = S (size' p)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index,
  size' (svar_open k n p) = size' p
k: db_index
S (size' p^[svar:S k↦patt_free_svar n]) = S (size' p)
  rewrite (IHp1 k); rewrite (IHp2 k); reflexivity.
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index,
  size' (svar_open k n p) = size' p
k: db_index
S (size' p^[svar:k↦patt_free_svar n]) = S (size' p)
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index,
  size' (svar_open k n p) = size' p
k: db_index
S (size' p^[svar:S k↦patt_free_svar n]) = S (size' p)
  rewrite (IHp k); reflexivity.
Σ: Signature
n: svar
p: Pattern
IHp: ∀ k : db_index,
  size' (svar_open k n p) = size' p
k: db_index
S (size' p^[svar:S k↦patt_free_svar n]) = S (size' p)
  rewrite (IHp (S k)); reflexivity.
Qed.

Check fold_left.
fold_left
     : ∀ A B : Type, (A → B → A) → list B → A → A
Definition bcmcloseex
    {Σ : Signature}
    (l : list (prod db_index evar))
    (ϕ : Pattern) : Pattern
:= fold_left (λ ϕ' p, evar_open p.1 p.2 ϕ') l ϕ.

Lemma bcmcloseex_append {Σ : Signature} (l₁ l₂ : list (prod db_index evar)) (ϕ : Pattern) :
  bcmcloseex (l₁ ++ l₂) ϕ = bcmcloseex l₂ (bcmcloseex l₁ ϕ).
Σ: Signature
l₁, l₂: list (db_index * evar)
ϕ: Pattern
bcmcloseex (l₁ ++ l₂) ϕ =
bcmcloseex l₂ (bcmcloseex l₁ ϕ)
Proof.
Σ: Signature
l₁, l₂: list (db_index * evar)
ϕ: Pattern
bcmcloseex (l₁ ++ l₂) ϕ =
bcmcloseex l₂ (bcmcloseex l₁ ϕ)
  unfold bcmcloseex.
Σ: Signature
l₁, l₂: list (db_index * evar)
ϕ: Pattern
fold_left
  (λ (ϕ' : Pattern) (p : db_index * evar),
     evar_open p.1 p.2 ϕ') (l₁ ++ l₂) ϕ =
fold_left
  (λ (ϕ' : Pattern) (p : db_index * evar),
     evar_open p.1 p.2 ϕ') l₂
  (fold_left
     (λ (ϕ' : Pattern) (p : db_index * evar),
        evar_open p.1 p.2 ϕ') l₁ ϕ) rewrite fold_left_app.
Σ: Signature
l₁, l₂: list (db_index * evar)
ϕ: Pattern
fold_left
  (λ (ϕ' : Pattern) (p : db_index * evar),
     evar_open p.1 p.2 ϕ') l₂
  (fold_left
     (λ (ϕ' : Pattern) (p : db_index * evar),
        evar_open p.1 p.2 ϕ') l₁ ϕ) =
fold_left
  (λ (ϕ' : Pattern) (p : db_index * evar),
     evar_open p.1 p.2 ϕ') l₂
  (fold_left
     (λ (ϕ' : Pattern) (p : db_index * evar),
        evar_open p.1 p.2 ϕ') l₁ ϕ) reflexivity.
Qed.

(*
Lemma bmcloseex_wfcex
  {Σ : Signature}
  (l : list (prod db_index evar))
  (ϕ : Pattern)
  : well_formed_closed_ex_aux ϕ from ->
  (bcmcloseex from l ϕ) = ϕ.
Proof.
  intros H.
  induction l.
  { reflexivity. }
  { simpl. rewrite IHl. by rewrite evar_open_not_occur. }
Qed.
*)
Lemma bcmcloseex_bott
  {Σ : Signature}
  (l : list (prod db_index evar))
  : bcmcloseex l patt_bott = patt_bott.
Σ: Signature
l: list (db_index * evar)
bcmcloseex l patt_bott = patt_bott
Proof.
Σ: Signature
l: list (db_index * evar)
bcmcloseex l patt_bott = patt_bott
  induction l.
Σ: Signature
bcmcloseex [] patt_bott = patt_bott
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: bcmcloseex l patt_bott = patt_bott
bcmcloseex (a :: l) patt_bott = patt_bott
  {
Σ: Signature
bcmcloseex [] patt_bott = patt_bott reflexivity. }
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: bcmcloseex l patt_bott = patt_bott
bcmcloseex (a :: l) patt_bott = patt_bott
  {
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: bcmcloseex l patt_bott = patt_bott
bcmcloseex (a :: l) patt_bott = patt_bott simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: bcmcloseex l patt_bott = patt_bott
bcmcloseex l (evar_open a.1 a.2 patt_bott) = patt_bott rewrite IHl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: bcmcloseex l patt_bott = patt_bott
patt_bott = patt_bott reflexivity. }
Qed.

Lemma bcmcloseex_sym
  {Σ : Signature}
  (l : list (prod db_index evar))
  (s : symbols)
  : bcmcloseex l (patt_sym s) = (patt_sym s).
Σ: Signature
l: list (db_index * evar)
s: symbols
bcmcloseex l (patt_sym s) = patt_sym s
Proof.
Σ: Signature
l: list (db_index * evar)
s: symbols
bcmcloseex l (patt_sym s) = patt_sym s
  induction l.
Σ: Signature
s: symbols
bcmcloseex [] (patt_sym s) = patt_sym s
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
s: symbols
IHl: bcmcloseex l (patt_sym s) = patt_sym s
bcmcloseex (a :: l) (patt_sym s) = patt_sym s
  {
Σ: Signature
s: symbols
bcmcloseex [] (patt_sym s) = patt_sym s reflexivity. }
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
s: symbols
IHl: bcmcloseex l (patt_sym s) = patt_sym s
bcmcloseex (a :: l) (patt_sym s) = patt_sym s
  {
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
s: symbols
IHl: bcmcloseex l (patt_sym s) = patt_sym s
bcmcloseex (a :: l) (patt_sym s) = patt_sym s simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
s: symbols
IHl: bcmcloseex l (patt_sym s) = patt_sym s
bcmcloseex l (evar_open a.1 a.2 (patt_sym s)) =
patt_sym s rewrite IHl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
s: symbols
IHl: bcmcloseex l (patt_sym s) = patt_sym s
patt_sym s = patt_sym s reflexivity. }
Qed.

Lemma bcmcloseex_imp
  {Σ : Signature}
  (l : list (prod db_index evar))
  (p q : Pattern)
  : bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q).
Σ: Signature
l: list (db_index * evar)
p, q: Pattern
bcmcloseex l (patt_imp p q) =
patt_imp (bcmcloseex l p) (bcmcloseex l q)
Proof.
Σ: Signature
l: list (db_index * evar)
p, q: Pattern
bcmcloseex l (patt_imp p q) =
patt_imp (bcmcloseex l p) (bcmcloseex l q)
  move: p q.
Σ: Signature
l: list (db_index * evar)
∀ p q : Pattern,
  bcmcloseex l (patt_imp p q) =
  patt_imp (bcmcloseex l p) (bcmcloseex l q)
  induction l; intros p q.
Σ: Signature
p, q: Pattern
bcmcloseex [] (patt_imp p q) =
patt_imp (bcmcloseex [] p) (bcmcloseex [] q)
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex (a :: l) (patt_imp p q) =
patt_imp (bcmcloseex (a :: l) p)
  (bcmcloseex (a :: l) q)
  {
Σ: Signature
p, q: Pattern
bcmcloseex [] (patt_imp p q) =
patt_imp (bcmcloseex [] p) (bcmcloseex [] q) reflexivity. }
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex (a :: l) (patt_imp p q) =
patt_imp (bcmcloseex (a :: l) p)
  (bcmcloseex (a :: l) q)
  {
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex (a :: l) (patt_imp p q) =
patt_imp (bcmcloseex (a :: l) p)
  (bcmcloseex (a :: l) q) simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex l (evar_open a.1 a.2 (patt_imp p q)) =
patt_imp (bcmcloseex l (evar_open a.1 a.2 p))
  (bcmcloseex l (evar_open a.1 a.2 q)) unfold evar_open.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex l
  (patt_imp p q)^[evar:a.1↦patt_free_evar a.2] =
patt_imp
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex l
  (patt_imp p^[evar:a.1↦patt_free_evar a.2]
     q^[evar:a.1↦patt_free_evar a.2]) =
patt_imp
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) rewrite IHl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_imp p q) = patt_imp (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
patt_imp
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) =
patt_imp
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) reflexivity. }
Qed.

Lemma bcmcloseex_app
  {Σ : Signature}
  (l : list (prod db_index evar))
  (p q : Pattern)
  : bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q).
Σ: Signature
l: list (db_index * evar)
p, q: Pattern
bcmcloseex l (patt_app p q) =
patt_app (bcmcloseex l p) (bcmcloseex l q)
Proof.
Σ: Signature
l: list (db_index * evar)
p, q: Pattern
bcmcloseex l (patt_app p q) =
patt_app (bcmcloseex l p) (bcmcloseex l q)
  move: p q.
Σ: Signature
l: list (db_index * evar)
∀ p q : Pattern,
  bcmcloseex l (patt_app p q) =
  patt_app (bcmcloseex l p) (bcmcloseex l q)
  induction l; intros p q.
Σ: Signature
p, q: Pattern
bcmcloseex [] (patt_app p q) =
patt_app (bcmcloseex [] p) (bcmcloseex [] q)
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex (a :: l) (patt_app p q) =
patt_app (bcmcloseex (a :: l) p)
  (bcmcloseex (a :: l) q)
  {
Σ: Signature
p, q: Pattern
bcmcloseex [] (patt_app p q) =
patt_app (bcmcloseex [] p) (bcmcloseex [] q) reflexivity. }
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex (a :: l) (patt_app p q) =
patt_app (bcmcloseex (a :: l) p)
  (bcmcloseex (a :: l) q)
  {
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex (a :: l) (patt_app p q) =
patt_app (bcmcloseex (a :: l) p)
  (bcmcloseex (a :: l) q) simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex l (evar_open a.1 a.2 (patt_app p q)) =
patt_app (bcmcloseex l (evar_open a.1 a.2 p))
  (bcmcloseex l (evar_open a.1 a.2 q)) unfold evar_open.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex l
  (patt_app p q)^[evar:a.1↦patt_free_evar a.2] =
patt_app
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
bcmcloseex l
  (patt_app p^[evar:a.1↦patt_free_evar a.2]
     q^[evar:a.1↦patt_free_evar a.2]) =
patt_app
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) rewrite IHl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ p q : Pattern,
bcmcloseex l (patt_app p q) = patt_app (bcmcloseex l p) (bcmcloseex l q)
p, q: Pattern
patt_app
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) =
patt_app
  (bcmcloseex l p^[evar:a.1↦patt_free_evar a.2])
  (bcmcloseex l q^[evar:a.1↦patt_free_evar a.2]) reflexivity. }
Qed.

Lemma bcmcloseex_ex
  {Σ : Signature}
  (l : list (prod db_index evar))
  (q : Pattern)
  : bcmcloseex l (patt_exists q) = patt_exists (bcmcloseex (map (λ p, (S p.1,p.2)) l) q).
Σ: Signature
l: list (db_index * evar)
q: Pattern
bcmcloseex l (patt_exists q) =
patt_exists
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     q)
Proof.
Σ: Signature
l: list (db_index * evar)
q: Pattern
bcmcloseex l (patt_exists q) =
patt_exists
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     q)
  move: q.
Σ: Signature
l: list (db_index * evar)
∀ q : Pattern,
  bcmcloseex l (patt_exists q) =
  patt_exists
    (bcmcloseex
       (map (λ p : nat * evar, (S p.1, p.2)) l) q)
  induction l; intros q.
Σ: Signature
q: Pattern
bcmcloseex [] (patt_exists q) =
patt_exists
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) []) q)
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern,
bcmcloseex l (patt_exists q) =
patt_exists (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) q)
q: Pattern
bcmcloseex (a :: l) (patt_exists q) =
patt_exists
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) (a :: l)) q)
  {
Σ: Signature
q: Pattern
bcmcloseex [] (patt_exists q) =
patt_exists
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) []) q) reflexivity. }
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern,
bcmcloseex l (patt_exists q) =
patt_exists (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) q)
q: Pattern
bcmcloseex (a :: l) (patt_exists q) =
patt_exists
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) (a :: l)) q)
  {
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern,
bcmcloseex l (patt_exists q) =
patt_exists (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) q)
q: Pattern
bcmcloseex (a :: l) (patt_exists q) =
patt_exists
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) (a :: l)) q) simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern,
bcmcloseex l (patt_exists q) =
patt_exists (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) q)
q: Pattern
bcmcloseex l (evar_open a.1 a.2 (patt_exists q)) =
patt_exists
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     (evar_open (S a.1) a.2 q)) rewrite IHl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern,
bcmcloseex l (patt_exists q) =
patt_exists (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) q)
q: Pattern
patt_exists
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     ((fix bevar_subst
         (phi psi : Pattern) (x : db_index) {struct
          phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | patt_app phi1 phi2 =>
             patt_app (bevar_subst phi1 psi x)
               (bevar_subst phi2 psi x)
         | patt_bott => patt_bott
         | patt_imp phi1 phi2 =>
             patt_imp (bevar_subst phi1 psi x)
               (bevar_subst phi2 psi x)
         | patt_exists phi' =>
             patt_exists (bevar_subst phi' psi (S x))
         | patt_mu phi' =>
             patt_mu (bevar_subst phi' psi x)
         end) q (patt_free_evar a.2) (S a.1))) =
patt_exists
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     (evar_open (S a.1) a.2 q)) reflexivity. }
Qed.

Lemma bcmcloseex_mu
  {Σ : Signature}
  (l : list (prod db_index evar))
  (q : Pattern)
  : bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q).
Σ: Signature
l: list (db_index * evar)
q: Pattern
bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
Proof.
Σ: Signature
l: list (db_index * evar)
q: Pattern
bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
  move: q.
Σ: Signature
l: list (db_index * evar)
∀ q : Pattern,
  bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
  induction l; intros q.
Σ: Signature
q: Pattern
bcmcloseex [] (patt_mu q) = patt_mu (bcmcloseex [] q)
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern, bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
q: Pattern
bcmcloseex (a :: l) (patt_mu q) =
patt_mu (bcmcloseex (a :: l) q)
  {
Σ: Signature
q: Pattern
bcmcloseex [] (patt_mu q) = patt_mu (bcmcloseex [] q) reflexivity. }
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern, bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
q: Pattern
bcmcloseex (a :: l) (patt_mu q) =
patt_mu (bcmcloseex (a :: l) q)
  {
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern, bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
q: Pattern
bcmcloseex (a :: l) (patt_mu q) =
patt_mu (bcmcloseex (a :: l) q) simpl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern, bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
q: Pattern
bcmcloseex l (evar_open a.1 a.2 (patt_mu q)) =
patt_mu (bcmcloseex l (evar_open a.1 a.2 q)) rewrite IHl.
Σ: Signature
a: (db_index * evar)%type
l: list (db_index * evar)
IHl: ∀ q : Pattern, bcmcloseex l (patt_mu q) = patt_mu (bcmcloseex l q)
q: Pattern
patt_mu
  (bcmcloseex l
     ((fix bevar_subst
         (phi psi : Pattern) (x : db_index) {struct
          phi} : Pattern :=
         match phi with
         | patt_free_evar x' => patt_free_evar x'
         | patt_free_svar x' => patt_free_svar x'
         | patt_bound_evar n =>
             match compare_nat n x with
             | Nat_less _ _ _ => patt_bound_evar n
             | Nat_equal _ _ _ => psi
             | Nat_greater _ _ _ =>
                 patt_bound_evar (Nat.pred n)
             end
         | patt_bound_svar n => patt_bound_svar n
         | patt_sym sigma => patt_sym sigma
         | patt_app phi1 phi2 =>
             patt_app (bevar_subst phi1 psi x)
               (bevar_subst phi2 psi x)
         | patt_bott => patt_bott
         | patt_imp phi1 phi2 =>
             patt_imp (bevar_subst phi1 psi x)
               (bevar_subst phi2 psi x)
         | patt_exists phi' =>
             patt_exists (bevar_subst phi' psi (S x))
         | patt_mu phi' =>
             patt_mu (bevar_subst phi' psi x)
         end) q (patt_free_evar a.2) a.1)) =
patt_mu (bcmcloseex l (evar_open a.1 a.2 q)) reflexivity. }
Qed.

Lemma wfc_ex_aux_S_bevar_subst_fe {Σ : Signature} k ϕ x:
  well_formed_closed_ex_aux ϕ^[evar:k↦patt_free_evar x] k = true ->
  well_formed_closed_ex_aux ϕ (S k) = true.
Σ: Signature
k: db_index
ϕ: Pattern
x: evar
well_formed_closed_ex_aux ϕ^[evar:k↦patt_free_evar x]
  k = true → well_formed_closed_ex_aux ϕ (S k) = true  
Proof.
Σ: Signature
k: db_index
ϕ: Pattern
x: evar
well_formed_closed_ex_aux ϕ^[evar:k↦patt_free_evar x]
  k = true → well_formed_closed_ex_aux ϕ (S k) = true
  intros H.
Σ: Signature
k: db_index
ϕ: Pattern
x: evar
H: well_formed_closed_ex_aux
  ϕ^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S k) = true move: k H.
Σ: Signature
ϕ: Pattern
x: evar
∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ (S k) = true
  induction ϕ; intros k H; simpl in *; auto.
Σ: Signature
n: db_index
x: evar
k: db_index
H: well_formed_closed_ex_aux
  match compare_nat n k with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end k = true
(if decide (n < S k) then true else false) = true
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true
  {
Σ: Signature
n: db_index
x: evar
k: db_index
H: well_formed_closed_ex_aux
  match compare_nat n k with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end k = true
(if decide (n < S k) then true else false) = true repeat case_match; auto; try lia.
Σ: Signature
n: db_index
x: evar
k: db_index
g: n > k
H0: compare_nat n k = Nat_greater n k g
H: well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) k = true
n0: ¬ n < S k
H1: decide (n < S k) = right n0
false = true simpl in H.
Σ: Signature
n: db_index
x: evar
k: db_index
g: n > k
H0: compare_nat n k = Nat_greater n k g
H: (if decide (Nat.pred n < k) then true else false) =
true
n0: ¬ n < S k
H1: decide (n < S k) = right n0
false = true case_match; lia. }
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true
  {
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true destruct_and!.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H0: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k = true
H1: well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true rewrite IHϕ1;[assumption|].
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H0: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k = true
H1: well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
true && well_formed_closed_ex_aux ϕ2 (S k) = true rewrite IHϕ2;[assumption|].
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H0: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k = true
H1: well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
true && true = true reflexivity. }
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true
  {
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true destruct_and!.
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H0: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k = true
H1: well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k) = true rewrite IHϕ1;[assumption|].
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H0: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k = true
H1: well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
true && well_formed_closed_ex_aux ϕ2 (S k) = true rewrite IHϕ2;[assumption|].
Σ: Signature
ϕ1, ϕ2: Pattern
x: evar
IHϕ1: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ1^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ1 (S k) = true
IHϕ2: ∀ k : db_index,
  well_formed_closed_ex_aux
    ϕ2^[evar:k↦patt_free_evar x] k = true
  → well_formed_closed_ex_aux ϕ2 (S k) = true
k: db_index
H0: well_formed_closed_ex_aux
  ϕ1^[evar:k↦patt_free_evar x] k = true
H1: well_formed_closed_ex_aux
  ϕ2^[evar:k↦patt_free_evar x] k = true
true && true = true reflexivity. }
Qed.

Lemma wfc_ex_aux_evar_open_gt {Σ : Signature} dbi x k ϕ:
  k > dbi ->
  well_formed_closed_ex_aux (evar_open dbi x ϕ) k ->
  well_formed_closed_ex_aux ϕ (S k).
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
k > dbi
→ well_formed_closed_ex_aux (evar_open dbi x ϕ) k
  → well_formed_closed_ex_aux ϕ (S k)
Proof.
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
k > dbi
→ well_formed_closed_ex_aux (evar_open dbi x ϕ) k
  → well_formed_closed_ex_aux ϕ (S k)
  unfold evar_open.
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
k > dbi
→ well_formed_closed_ex_aux
    ϕ^[evar:dbi↦patt_free_evar x] k
  → well_formed_closed_ex_aux ϕ (S k)
  intros H1 H2.
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
  move: k dbi H1 H2.
Σ: Signature
x: evar
ϕ: Pattern
∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
  induction ϕ; intros k dbi H1 H2; simpl in *; auto.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end k
if decide (n < S k) then true else false
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k)
well_formed_closed_ex_aux ϕ (S (S k))
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
  {
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end k
if decide (n < S k) then true else false 
    repeat case_match; simpl; repeat case_match; auto; try lia.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k > dbi
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H2: well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) k
n0: ¬ n < S k
H0: decide (n < S k) = right n0
false
    simpl in H2.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k > dbi
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H2: if decide (Nat.pred n < k) then true else false
n0: ¬ n < S k
H0: decide (n < S k) = right n0
false case_match; try lia.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k > dbi
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
n1: ¬ Nat.pred n < k
H3: decide (Nat.pred n < k) = right n1
H2: false
n0: ¬ n < S k
H0: decide (n < S k) = right n0
false apply H2.
  }
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k)
well_formed_closed_ex_aux ϕ (S (S k))
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
  {
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
    destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
    rewrite (IHϕ1 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && well_formed_closed_ex_aux ϕ2 (S k)
    rewrite (IHϕ2 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && true
    reflexivity.
  }
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k)
well_formed_closed_ex_aux ϕ (S (S k))
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
  {
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
    destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S k) &&
well_formed_closed_ex_aux ϕ2 (S k)
    rewrite (IHϕ1 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && well_formed_closed_ex_aux ϕ2 (S k)
    rewrite (IHϕ2 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ1^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ1 (S k)
IHϕ2: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ2^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ2 (S k)
k, dbi: nat
H1: k > dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && true
    reflexivity.
  }
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k)
well_formed_closed_ex_aux ϕ (S (S k))
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
  {
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k)
well_formed_closed_ex_aux ϕ (S (S k))
    rewrite (IHϕ (S k) (S dbi));[lia|assumption|].
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k)
true
    reflexivity.
  }
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
  {
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
well_formed_closed_ex_aux ϕ (S k)
    eapply IHϕ;[|eassumption].
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k > dbi
  → well_formed_closed_ex_aux
      ϕ^[evar:dbi↦patt_free_evar x] k
    → well_formed_closed_ex_aux ϕ (S k)
k, dbi: nat
H1: k > dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k
k > dbi lia.
  }
Qed.

Lemma wfc_ex_aux_evar_open_lt {Σ : Signature} dbi x k ϕ:
  k < dbi ->
  well_formed_closed_ex_aux (evar_open dbi x ϕ) k = true ->
  well_formed_closed_ex_aux ϕ (S dbi) = true.
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
k < dbi
→ well_formed_closed_ex_aux (evar_open dbi x ϕ) k =
  true → well_formed_closed_ex_aux ϕ (S dbi) = true
Proof.
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
k < dbi
→ well_formed_closed_ex_aux (evar_open dbi x ϕ) k =
  true → well_formed_closed_ex_aux ϕ (S dbi) = true
  intros H1 H2.
Σ: Signature
dbi: nat
x: evar
k: nat
ϕ: Pattern
H1: k < dbi
H2: well_formed_closed_ex_aux (evar_open dbi x ϕ) k =
true
well_formed_closed_ex_aux ϕ (S dbi) = true
  move: k dbi H1 H2.
Σ: Signature
x: evar
ϕ: Pattern
∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux (evar_open dbi x ϕ) k =
    true → well_formed_closed_ex_aux ϕ (S dbi) = true
  induction ϕ; intros k dbi H1 H2; simpl in *; auto.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  (evar_open dbi x (patt_bound_evar n)) k = true
(if decide (n < S dbi) then true else false) = true
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ1) k = true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ2) k = true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ1) k = true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ2) k = true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ) k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k) = true
well_formed_closed_ex_aux ϕ (S (S dbi)) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ) k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S dbi) = true
  {
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  (evar_open dbi x (patt_bound_evar n)) k = true
(if decide (n < S dbi) then true else false) = true
    unfold evar_open in H2.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  (patt_bound_evar n)^[evar:dbi↦
  patt_free_evar x] k = true
(if decide (n < S dbi) then true else false) = true simpl in H2.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  match compare_nat n dbi with
  | Nat_less _ _ _ => patt_bound_evar n
  | Nat_equal _ _ _ => patt_free_evar x
  | Nat_greater _ _ _ =>
      patt_bound_evar (Nat.pred n)
  end k = true
(if decide (n < S dbi) then true else false) = true repeat case_match; auto; try lia.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k < dbi
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H2: well_formed_closed_ex_aux
  (patt_bound_evar (Nat.pred n)) k = true
n0: ¬ n < S dbi
H0: decide (n < S dbi) = right n0
false = true
    simpl in H2.
Σ: Signature
x: evar
n: db_index
k, dbi: nat
H1: k < dbi
g: n > dbi
H: compare_nat n dbi = Nat_greater n dbi g
H2: (if decide (Nat.pred n < k) then true else false) =
true
n0: ¬ n < S dbi
H0: decide (n < S dbi) = right n0
false = true case_match; lia.
  }
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ1) k = true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ2) k = true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ1) k = true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ2) k = true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ) k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k) = true
well_formed_closed_ex_aux ϕ (S (S dbi)) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ) k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S dbi) = true
  {
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ1) k = true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ2) k = true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
    destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ1) k = true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ2) k = true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
    rewrite (IHϕ1 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ1) k = true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ2) k = true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && well_formed_closed_ex_aux ϕ2 (S dbi) = true
    rewrite (IHϕ2 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ1) k = true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ2) k = true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && true = true
    reflexivity.
  }
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ1) k = true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ2) k = true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ) k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k) = true
well_formed_closed_ex_aux ϕ (S (S dbi)) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ) k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S dbi) = true
  {
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ1) k = true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ2) k = true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k &&
well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
    destruct_and!.
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ1) k = true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ2) k = true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ1 (S dbi) &&
well_formed_closed_ex_aux ϕ2 (S dbi) = true
    rewrite (IHϕ1 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ1) k = true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ2) k = true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && well_formed_closed_ex_aux ϕ2 (S dbi) = true
    rewrite (IHϕ2 k dbi);[assumption|assumption|].
Σ: Signature
x: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ1) k = true
    → well_formed_closed_ex_aux ϕ1 (S dbi) =
      true
IHϕ2: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ2) k = true
    → well_formed_closed_ex_aux ϕ2 (S dbi) =
      true
k, dbi: nat
H1: k < dbi
H: well_formed_closed_ex_aux
  ϕ1^[evar:dbi↦patt_free_evar x] k = true
H0: well_formed_closed_ex_aux
  ϕ2^[evar:dbi↦patt_free_evar x] k = true
true && true = true
    reflexivity.
  }
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ) k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k) = true
well_formed_closed_ex_aux ϕ (S (S dbi)) = true
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ) k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S dbi) = true
  {
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ) k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k) = true
well_formed_closed_ex_aux ϕ (S (S dbi)) = true
    rewrite (IHϕ (S k) (S dbi));[lia|assumption|].
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ) k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:S dbi↦patt_free_evar x] 
  (S k) = true
true = true
    reflexivity.
  }
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ) k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S dbi) = true
  {
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ) k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
well_formed_closed_ex_aux ϕ (S dbi) = true
    eapply IHϕ;[|eassumption].
Σ: Signature
x: evar
ϕ: Pattern
IHϕ: ∀ k dbi : nat,
  k < dbi
  → well_formed_closed_ex_aux
      (evar_open dbi x ϕ) k = true
    → well_formed_closed_ex_aux ϕ (S dbi) = true
k, dbi: nat
H1: k < dbi
H2: well_formed_closed_ex_aux
  ϕ^[evar:dbi↦patt_free_evar x] k = true
k < dbi lia.
  }
Qed.

Lemma evar_open_twice_not_occur {Σ : Signature} n x y ϕ:
  bevar_occur ϕ n = false ->
  evar_open n y (evar_open n x ϕ) = evar_open n x (evar_open (S n) y ϕ).
Σ: Signature
n: db_index
x, y: evar
ϕ: Pattern
bevar_occur ϕ n = false
→ evar_open n y (evar_open n x ϕ) =
  evar_open n x (evar_open (S n) y ϕ)
Proof.
Σ: Signature
n: db_index
x, y: evar
ϕ: Pattern
bevar_occur ϕ n = false
→ evar_open n y (evar_open n x ϕ) =
  evar_open n x (evar_open (S n) y ϕ)
  unfold evar_open.
Σ: Signature
n: db_index
x, y: evar
ϕ: Pattern
bevar_occur ϕ n = false
→ ϕ^[evar:n↦patt_free_evar x]^[evar:n↦patt_free_evar y] =
  ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦patt_free_evar
                                          x]
  move: n.
Σ: Signature
x, y: evar
ϕ: Pattern
∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦patt_free_evar
                                          y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦patt_free_evar
                                            x]
  induction ϕ; intros n' Hnoc; simpl in *; auto.
Σ: Signature
x, y: evar
n, n': db_index
Hnoc: (if decide (n = n') then true else false) =
false
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:n'↦patt_free_evar y] =
match compare_nat n (S n') with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar y
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:n'↦patt_free_evar x]
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ (S n') = false
patt_exists
  ϕ^[evar:S n'↦patt_free_evar x]^[evar:
  S n'↦patt_free_evar y] =
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:
  S n'↦patt_free_evar x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  {
Σ: Signature
x, y: evar
n, n': db_index
Hnoc: (if decide (n = n') then true else false) =
false
match compare_nat n n' with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar x
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:n'↦patt_free_evar y] =
match compare_nat n (S n') with
| Nat_less _ _ _ => patt_bound_evar n
| Nat_equal _ _ _ => patt_free_evar y
| Nat_greater _ _ _ => patt_bound_evar (Nat.pred n)
end^[evar:n'↦patt_free_evar x]
    repeat case_match; simpl; auto; try lia; repeat case_match; auto; try congruence; lia.
  }
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ (S n') = false
patt_exists
  ϕ^[evar:S n'↦patt_free_evar x]^[evar:
  S n'↦patt_free_evar y] =
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:
  S n'↦patt_free_evar x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  {
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
    apply orb_false_elim in Hnoc.
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' = false
∧ bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
    destruct_and!.
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
H: bevar_occur ϕ1 n' = false
H0: bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
    rewrite (IHϕ1 n'); [assumption|].
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
H: bevar_occur ϕ1 n' = false
H0: bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
    rewrite (IHϕ2 n'); [assumption|].
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
H: bevar_occur ϕ1 n' = false
H0: bevar_occur ϕ2 n' = false
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x] =
patt_app
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
    reflexivity.
  }
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ (S n') = false
patt_exists
  ϕ^[evar:S n'↦patt_free_evar x]^[evar:
  S n'↦patt_free_evar y] =
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:
  S n'↦patt_free_evar x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  {
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' || bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
    apply orb_false_elim in Hnoc.
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ1 n' = false
∧ bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
    destruct_and!.
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
H: bevar_occur ϕ1 n' = false
H0: bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
    rewrite (IHϕ1 n'); [assumption|].
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
H: bevar_occur ϕ1 n' = false
H0: bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
    rewrite (IHϕ2 n'); [assumption|].
Σ: Signature
x, y: evar
ϕ1, ϕ2: Pattern
IHϕ1: ∀ n : db_index,
  bevar_occur ϕ1 n = false
  → ϕ1^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ1^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
IHϕ2: ∀ n : db_index,
  bevar_occur ϕ2 n = false
  → ϕ2^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ2^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
H: bevar_occur ϕ1 n' = false
H0: bevar_occur ϕ2 n' = false
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x] =
patt_imp
  ϕ1^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  ϕ2^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
    reflexivity.
  }
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ (S n') = false
patt_exists
  ϕ^[evar:S n'↦patt_free_evar x]^[evar:
  S n'↦patt_free_evar y] =
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:
  S n'↦patt_free_evar x]
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  {
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ (S n') = false
patt_exists
  ϕ^[evar:S n'↦patt_free_evar x]^[evar:
  S n'↦patt_free_evar y] =
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:
  S n'↦patt_free_evar x]
    rewrite (IHϕ (S n'));[assumption|].
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ (S n') = false
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:
  S n'↦patt_free_evar x] =
patt_exists
  ϕ^[evar:S (S n')↦patt_free_evar y]^[evar:
  S n'↦patt_free_evar x]
    reflexivity.
  }
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
  {
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:n'↦patt_free_evar x]^[evar:n'↦
  patt_free_evar y] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
    rewrite IHϕ;[assumption|].
Σ: Signature
x, y: evar
ϕ: Pattern
IHϕ: ∀ n : db_index,
  bevar_occur ϕ n = false
  → ϕ^[evar:n↦patt_free_evar x]^[evar:n↦
    patt_free_evar y] =
    ϕ^[evar:S n↦patt_free_evar y]^[evar:n↦
    patt_free_evar x]
n': db_index
Hnoc: bevar_occur ϕ n' = false
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x] =
patt_mu
  ϕ^[evar:S n'↦patt_free_evar y]^[evar:n'↦
  patt_free_evar x]
    reflexivity.
  }
Qed.

Lemma wfc_ex_aux_bcmcloseex {Σ : Signature} l k ϕ:
  Forall (λ p : nat * evar, p.1 ≤ k) l ->
  well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true ->
  well_formed_closed_ex_aux (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true.
Σ: Signature
l: list (nat * evar)
k: nat
ϕ: Pattern
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux
    (bcmcloseex l (patt_exists ϕ)) k = true
  → well_formed_closed_ex_aux
      (bcmcloseex
         (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ)
      (S k) = true
Proof.
Σ: Signature
l: list (nat * evar)
k: nat
ϕ: Pattern
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux
    (bcmcloseex l (patt_exists ϕ)) k = true
  → well_formed_closed_ex_aux
      (bcmcloseex
         (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ)
      (S k) = true
  intros Hk H.
Σ: Signature
l: list (nat * evar)
k: nat
ϕ: Pattern
Hk: Forall (λ p : nat * evar, p.1 ≤ k) l
H: well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists ϕ)) k = true
well_formed_closed_ex_aux
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     ϕ) (S k) = true
  move: ϕ k Hk H.
Σ: Signature
l: list (nat * evar)
∀ (ϕ : Pattern) (k : nat),
  Forall (λ p : nat * evar, p.1 ≤ k) l
  → well_formed_closed_ex_aux
      (bcmcloseex l (patt_exists ϕ)) k = true
    → well_formed_closed_ex_aux
        (bcmcloseex
           (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ)
        (S k) = true
  induction l; intros ϕ k Hk H.
Σ: Signature
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) []
H: well_formed_closed_ex_aux
  (bcmcloseex [] (patt_exists ϕ)) k = true
well_formed_closed_ex_aux
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) []) ϕ)
  (S k) = true
Σ: Signature
a: (nat * evar)%type
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) (a :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex (a :: l) (patt_exists ϕ)) k = true
well_formed_closed_ex_aux
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) (a :: l)) ϕ)
  (S k) = true
  {
Σ: Signature
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) []
H: well_formed_closed_ex_aux
  (bcmcloseex [] (patt_exists ϕ)) k = true
well_formed_closed_ex_aux
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) []) ϕ)
  (S k) = true simpl.
Σ: Signature
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) []
H: well_formed_closed_ex_aux
  (bcmcloseex [] (patt_exists ϕ)) k = true
well_formed_closed_ex_aux ϕ (S k) = true simpl in H.
Σ: Signature
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) []
H: well_formed_closed_ex_aux ϕ (S k) = true
well_formed_closed_ex_aux ϕ (S k) = true apply H. }
Σ: Signature
a: (nat * evar)%type
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) (a :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex (a :: l) (patt_exists ϕ)) k = true
well_formed_closed_ex_aux
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) (a :: l)) ϕ)
  (S k) = true
  {
Σ: Signature
a: (nat * evar)%type
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k) (a :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex (a :: l) (patt_exists ϕ)) k = true
well_formed_closed_ex_aux
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2)) (a :: l)) ϕ)
  (S k) = true
    destruct a as [dbi x].
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex ((dbi, x) :: l) (patt_exists ϕ)) k =
true
well_formed_closed_ex_aux
  (bcmcloseex
     (map (λ p : nat * evar, (S p.1, p.2))
        ((dbi, x) :: l)) ϕ) (S k) = true
    simpl.
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex ((dbi, x) :: l) (patt_exists ϕ)) k =
true
well_formed_closed_ex_aux
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     (evar_open (S dbi) x ϕ)) (S k) = true simpl in H.
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (evar_open dbi x (patt_exists ϕ)))
  k = true
well_formed_closed_ex_aux
  (bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l)
     (evar_open (S dbi) x ϕ)) (S k) = true
    apply IHl.
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (evar_open dbi x (patt_exists ϕ)))
  k = true
Forall (λ p : nat * evar, p.1 ≤ k) l
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (evar_open dbi x (patt_exists ϕ)))
  k = true
well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists (evar_open (S dbi) x ϕ)))
  k = true
    {
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (evar_open dbi x (patt_exists ϕ)))
  k = true
Forall (λ p : nat * evar, p.1 ≤ k) l inversion Hk.
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (evar_open dbi x (patt_exists ϕ)))
  k = true
x0: (nat * evar)%type
l0: list (nat * evar)
H2: (dbi, x).1 ≤ k
H3: Forall (λ p : nat * evar, p.1 ≤ k) l
H0: x0 = (dbi, x)
H1: l0 = l
Forall (λ p : nat * evar, p.1 ≤ k) l subst.
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (evar_open dbi x (patt_exists ϕ)))
  k = true
H2: (dbi, x).1 ≤ k
H3: Forall (λ p : nat * evar, p.1 ≤ k) l
Forall (λ p : nat * evar, p.1 ≤ k) l assumption. }
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (evar_open dbi x (patt_exists ϕ)))
  k = true
well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists (evar_open (S dbi) x ϕ)))
  k = true
    {
Σ: Signature
dbi: nat
x: evar
l: list (nat * evar)
IHl: ∀ (ϕ : Pattern) (k : nat),
Forall (λ p : nat * evar, p.1 ≤ k) l
→ well_formed_closed_ex_aux (bcmcloseex l (patt_exists ϕ)) k = true
→ well_formed_closed_ex_aux
(bcmcloseex (map (λ p : nat * evar, (S p.1, p.2)) l) ϕ) (S k) = true
ϕ: Pattern
k: nat
Hk: Forall (λ p : nat * evar, p.1 ≤ k)
  ((dbi, x) :: l)
H: well_formed_closed_ex_aux
  (bcmcloseex l (evar_open dbi x (patt_exists ϕ)))
  k = true
well_formed_closed_ex_aux
  (bcmcloseex l (patt_exists (evar_open (S dbi) x ϕ)))
  k = true apply H. }
  }
Qed.

Lemma free_svars_free_evar_subst {Σ : Signature} ϕ x ψ:
  free_svars (free_evar_subst ϕ ψ x) ⊆ free_svars ϕ ∪ free_svars ψ.
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
free_svars ϕ^[[evar:x↦ψ]]
⊆ free_svars ϕ ∪ free_svars ψ
Proof.
Σ: Signature
ϕ: Pattern
x: evar
ψ: Pattern
free_svars ϕ^[[evar:x↦ψ]]
⊆ free_svars ϕ ∪ free_svars ψ
  induction ϕ; simpl; try set_solver.
Σ: Signature
x0, x: evar
ψ: Pattern
free_svars
  (if decide (x = x0) then ψ else patt_free_evar x0)
⊆ ∅ ∪ free_svars ψ
  {
Σ: Signature
x0, x: evar
ψ: Pattern
free_svars
  (if decide (x = x0) then ψ else patt_free_evar x0)
⊆ ∅ ∪ free_svars ψ
    case_match.
Σ: Signature
x0, x: evar
ψ: Pattern
e: x = x0
H: decide (x = x0) = left e
free_svars ψ ⊆ ∅ ∪ free_svars ψ
Σ: Signature
x0, x: evar
ψ: Pattern
n: x ≠ x0
H: decide (x = x0) = right n
free_svars (patt_free_evar x0) ⊆ ∅ ∪ free_svars ψ
    {
Σ: Signature
x0, x: evar
ψ: Pattern
e: x = x0
H: decide (x = x0) = left e
free_svars ψ ⊆ ∅ ∪ free_svars ψ
      subst.
Σ: Signature
x0: evar
ψ: Pattern
H: decide (x0 = x0) = left (erefl x0)
free_svars ψ ⊆ ∅ ∪ free_svars ψ set_solver.
    }
Σ: Signature
x0, x: evar
ψ: Pattern
n: x ≠ x0
H: decide (x = x0) = right n
free_svars (patt_free_evar x0) ⊆ ∅ ∪ free_svars ψ
    {
Σ: Signature
x0, x: evar
ψ: Pattern
n: x ≠ x0
H: decide (x = x0) = right n
free_svars (patt_free_evar x0) ⊆ ∅ ∪ free_svars ψ
      simpl.
Σ: Signature
x0, x: evar
ψ: Pattern
n: x ≠ x0
H: decide (x = x0) = right n
∅ ⊆ ∅ ∪ free_svars ψ set_solver.
    }
  }
Qed.